Turbulence Models and Their Application: Efficient Numerical Methods With Computer Programs

-'" fill n.aao 0 () 5 0 TUNCFR CEBECI Turbulence Models and Their Application With Springer 1101(1;.)` C"OM H...

Author: Tuncer Cebeci

72 downloads 515 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

-'"

fill

n.aao

0

() 5

0

TUNCFR CEBECI

Turbulence Models and Their Application With

Springer 1101(1;.)`

C"OM

HORIZONS

Springer

PUBLISHING

Berlin Heidelberg

London Milan

New York

Paris

Hong Kong

Tokyo

Long Beach, California Heidelberg, Germany

;n::.

.:.:

:::.:.... ... .... .....::-.....: ...:.

n.

''

.R.Pfti^:.,;:;

52:., .. ..,..:.%':,y;.;s:::5:::;;:::aGr.':<,a:r:....::,:.,::::.,.. 'a.::':':.s...s<.,.....:...,f::::i:::.«,..,..,::,....::%::.....r:::.:e.::..:..x:..r.:,n<......'.c:.a::.:: a:.e.,..r,::: _......:,>..... y.:....

Tuncer Cebeci

Turbulence Models and Their Application Efficient Numerical Methods with Computer Programs

With 40 Figures and a CD-ROM

Springer HORIZONS PUBLISHING

Trotter Cebeci 810 Rancho Drive Long Beach, CA 90815, USA [email protected]

Library of Congress Cataloging-in-Publication Data Cebeci, Tuncer Turbulence models and their application: efficient numerical methods with computer programs/Tuncer Cebeci. p. cm. Includes bibliographical references and index. ISBN 0-9668461-6-8 (acid-free paper) 1. Turbulent boundary layer - Mathematical models. 2. Turbulence - Mathematical models. I. Title. QA9I3.C425 2003 532'.0527'015118-dc2l 2003014170

ISBN 0-9668461-6-8 Horizons Publishing Inc., Long Beach ISBN 3-540-40288-8 Springer-Verlag Berlin Heidelberg New York All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Horizons Publishing Inc., 810 Rancho Drive, Long Beach, CA 90815, USA) except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. © Horizons Publishing Inc., Long Beach. California 2004 Printed in Germany

The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Please note: All rights pertaining to the Computer Programs are owned exclusively by the author and Horizons Publishing Inc. The publisher and the author accept no legal responsibility for any damage caused by improper use of the programs. Although the programs have been tested with extreme care, errors cannot be excluded. Typeset in MS Word by the author. Edited and reformatted by Kurt Mattes, Heidelberg, Germany. using LATEX.

Cover design: Erich Kirchner, Heidelberg, Germany

Printed on acid-free paper 543210

Preface

Turbulence models are required to close forms of the time-independent NavierStokes equations, which involve unknown Reynolds stress correlations. The equations may be incompressible or compressible, two- or three-dimensional, appropriate to separated or boundary-layer flows and with correlations that involve from one shear stress to three shear stresses and three normal stresses: and the averaging may involve ensembles, time and density weighting. In all cases, the averaging implies more unknowns than equations and assumptions are necessary to close the set. An important consequence is that solutions of equations closed with turbulence models are no longer exact representations of physical problems implied by the boundary conditions, and the uncertainties associated with the assumptions have to be appraised, usually by comparison with experiments. Many models have been suggested over the last forty years, with initial emphasis on comparatively simple free flows and wall flows in which the unknown Reynolds-stress tensor could be reduced to one dominant shear-stress component. The ability to measure the shear stress and related mean-velocity gradients implied that assumptions could be evaluated directly and often, but not always, with encouraging results. It has proved more difficult to make evaluations of similar quality in more complex flows, for example those involving three independent variables and for substantial regions of flow separation, for which the Reynolds stress tensor involves more terms, although the correspondingly more complex models involve assumptions and empirical coefficients derived by reference to simpler flows. Thus, the extent to which a turbulence model is able to represent a flow depends on the complexity involved; assessments can be difficult and involve subjective judgement. At the same time, the Reynolds-stress tensor is one group of terms in the equation representing conservation of mean

momentum and its magnitude is, on occasion, small compared to the terms representing pressure gradients and sources. The implementation of turbulence models involves the numerical solution of the conservation equations for mass and momentum with the model assumptions for turbulent diffusion, and some compatibility between equations and assumptions is required to ensure solutions with reasonable computer resources. Thus, it can be expected that models in algebraic form are likely to allow results to be

Preface

VI

obtained with less cost, effort and time than those with models in the form of added differential equations for turbulence quantities, and some tradeoff may be required between cost and convenience on the one hand and greater generality on the other.

In this book, after a brief review of the more popular turbulence models (Chap. 1), we present and discuss accurate and efficient numerical methods for solving the boundary-layer equations with turbulence models based on algebraic formulas (mixing length, eddy viscosity) or partial-differential transport equations. In Chap. 2 we discuss the numerical solution of the boundary-layer

equations using the Cebeci-Smith model and the k-e model with and without wall functions, including a zonal method, all for flows without separation. A computer program employing the Cebeci-Smith model and the k-E model for obtaining the solution of two-dimensional incompressible turbulent flows is discussed in detail in Chap. 3 and is presented in the accompanying CD-ROM. The numerical solution of the boundary-layer equations with flow separation is discussed in Chap. 4. An inverse boundary-layer method employing the Cebeci-Smith turbulence model is described in detail. A computer program for obtaining boundary-layer solutions on airfoils and wakes is discussed and presented in the accompanying CD-ROM. This code, with Veldman's interaction law, can also be used interactively with the Hess and Smith panel method de-

scribed in Chap. 5. The panel method, also presented in the accompanying CD-ROM, includes the viscous effects. It is arranged in such a way that, with the computer program of Chap. 4, it can be used to obtain solutions of airfoil flows for a wide range of angles of attack, including stall, as discussed in Chap. 7. Chapter 6 discusses the application of the computer program for the Cebeci-Smith and k-e models to other higher-order turbulence models, including flows with separation. Finally, in Chap. 7 test cases are presented for the four companion computer programs in the accompanying CD-ROM. In many respects, this book can be seen as complementary and supplementary to that of Cebeci and Smith Analysis of Turbulent Boundary Layers (Academic Press, 1974) and its replacement Analysis of Turbulent Flows by the present author which is about to be published (Elsevier, 2003). It is a pleasure to acknowledge the help received from Dr. K. C. Chang, who read the whole manuscript and assisted with the development of the boundarylayer codes. Dr. J. P. Shao prepared the CD-ROM for the computer programs and Professor H. H. Chen provided help in the development of the panel code with viscous effects. Thanks are also due to Mr. Kurt Mattes who typed the manuscript and to Mr. Karl Koch who helped with the the production of the book.

Indian Wells, July. 2003

Tuncer Cebeci

;:r::. r

Contents

1.

Turbulence Models .........................................

I

Introduction .............................................

1

1.1 CS Model ................................................ 1.2 k-E Model ................................................

2

1.0

2.

5

k-w Model ...............................................

9

1.4 SST Model ...............................................

10

1.5 SA Model ................................................ References ...................................................

13

Solution of the CS and k-e Model Equations ................

17

2.0

Introduction .............................................

17

2.1

Transformed Variables .....................................

17

2.2

Solution Procedure ........................................

19

2.3

Solution of the k-E Equations with a Zonal Method ...........

22

2.3.1

Inner Region ......................................

22

2.3.2

Interface Between Inner and Outer Regions ............

27

2.3.3

Outer Region ......................................

28

2.3.4

Block-Elimination Method ..........................

30

1.3

14

2.4 Solution of the k-c Model Equations with and Without Wall Functions ...........................

2.4.1

Without Wall Functions .............................

2.4.2

31

Solution of the k-E Model Equations 31

Solution of the k-E Model Equations

with Wall Functions ................................

33

References ...................................................

34

Contents

VIII

3.

Computer Program for CS and k-e Models .................

Introduction .............................................

35 35

Common to Both Models ..................................

35

3.1.1

MAIN ............................................

35

3.1.2

Subroutine INPUT .................................

36

3.1.3

Subroutine IVPT ..................................

37

3.1.4

39

3.1.5

Subroutine GROWTH .............................. Subroutine GRID ..................................

3.1.6

Subroutine OUTPUT ...............................

40

Part 2 - CS Model ........................................

40

3.0 3.1 Part 1 - Components of the Computer Program

3.2

3.3

4.

5.

39

3.2.1

Subroutine COEFTR ...............................

41

3.2.2

Subroutine SOLV3 .................................

43

3.2.3

Subroutines EDDY, GAMCAL, CALFA ...............

45

Part 3 - k-e Model ........................................

46

3.3.1

Subroutines KECOEF, KEPARM, KEDEF

3.3.3

and KEDAMP ..................................... Subroutine KEINITK ............................... Subroutine KEINITG ...............................

46

3.3.2

3.3.4

Subroutine KEWALL ...............................

52

3.3.5

Subroutine KESOLV ...............................

52

3.4

Part 4 - Solution Algorithm ................................

52

3.5

Part 5 - Basic Tools .......................................

54

References ...................................................

54

Computer Program for CS Model for Flows with Separation

55

4.0

Introduction .............................................

55

4.1

Numerical Method ........................................

58

4.2

Computer Program 4.2.1 Subroutine Subroutine 4.2.2 4.2.3 Subroutine

....................................... INPUT ................................. IVPL ................................... HIC ....................................

62

References ...................................................

66

Hess-Smith Panel Method with Viscous Effects .............

67 67

5.0

Introduction .............................................

5.1

Mathematical Models for the Interaction Process

51 52

63 64 64

in Inviscid Flows ..........................................

68

5.2 HS Panel Method .........................................

71

5.3

Viscous Effects ...........................................

77

5.4

Flowfield Calculation in the Wake ...........................

5.5 Computer Program .......................................

80 82

MAIN ............................................

82

5.5.1

ix

Contents

5.5.2 5.5.3 5.5.4

Subroutine COEF ..................................

Subroutine OBKUTA ............................... Subroutine GAUSS .................................

5.5.5

Subroutine VPDIS .................................

5.5.6

Subroutine CLCM .................................. Subroutine VPDWK ................................

5.5.7

References ...................................................

6.

7.

Application of the Computer Program for the CS and k-e Models to Other Higher-Order Turbulence Models . 6.0

Introduction .............................................

6.1

Solution of the k-w and SST Model Equations ................

6.2 6.3

Solution of the SA Model Equations ......................... Extension of k-e Model to Flows with Separation .............

Companion Computer Programs ............................

83 84 84 84

85 85 85 88 94

Introduction ............................................. .......................

97 97 98

Without Viscous Effects ...................................

99

7.0 7.1 Test Cases for the CS and k-e Models 7.2 Sample Calculations for the Panel Method 7.3 7.4

82 82 . 83

Sample Calculations for the Inverse Boundary-Layer Program .. 103 Sample Calculations with the Interactive Boundary-Layer Program .................................. 105

7.5

Computer Programs in the CD-ROM ........................ 112

References ................................................... 116

Subject Index .................................................. 117

Turbulence Models

1.0 Introduction The use of Reynolds-averaged equations, made necessary by our inability to solve the time-dependent, three-dimensional Navier-Stokes equations with adequate resolution of time and spatial scales, see [1], implies that information has been lost and that further approximations are required to represent the fluctuating quantities known as Reynolds stresses and so reduce the number of unknowns to equal the number of equations. The most common approach to this problem is to define an eddy viscosity, e,,,,, in the same form as the laminar viscosity. Thus, for a two-dimensional incompressible flow,

- AuIvi = PE, uy

(1.0.1)

Another approach is to use the mixing length, t, concept and express the

Reynolds shear stress by

-

2()2

(1.0.2)

The specification of c,,, or e may be made in terms of algebraic equations or in terms of a combination of algebraic and differential equations and this has given rise to terminology involving the number of differential equations. Thus, the closures may be described in terms of zero, one and two differential equations. For a two-dimensional boundary-layer, the zero-equation approach usually treats a turbulent boundary layer as a composite layer with separate or a in each region. The Cebeci-Smith (CS) model discussed expressions for in detail in [1, 2] and briefly in Section 1.1 is a typical example for this approach. In the one-differential-equation approach the eddy viscosity- is written, with

c.. denoting a constant, as

1. Turbulence Models

2

Em = c,,0"22

(1.0.3)

with k obtained from a differential equation which represents the transport of turbulence energy and a from an algebraic formula. The Spalart-Allmaras (SA) model discussed in detail in [1, 3,4] and briefly in Section 1.5 is a good, useful model that uses this approach. In the two differential equation approach, the eddy viscosity is written as E.n, =

c, k2 (1.0.4)

e

with k and E obtained from differential equations which represent the transport of turbulence energy and its rate of dissipation. While one-equation models have found little favor except for the SA model, and where transport of turbulence characteristics is important as in strong adverse gradients or in separated flows, two equations have found extensive use. Various forms of two-equation models have been proposed and details have been given, for example in [1, 4]. Three popular models that are based on this approach are the k-E model discussed in Section 1.2 and the k-w and SST models briefly discussed in Sections 1.3 and 1.4, respectively. The Reynolds shear stress can also be modelled by using the Reynolds trans-

port equation as described for example in [1, 4]. A popular model is due to Launder, Reece and Rodi [5]. As pointed out by Bradshaw "it is so obvious that stress-transport models are more realistic in principle than eddy viscosity models that the improvements they give are very disappointing and most engineers have decided that the increased numerical difficulties (complexity of programming, expense of calculation, occasional instability) do not warrant changing up from eddy-viscosity models at present. Even stress-transport models often give very poor predictions of complex flows - notoriously, the effects of streamline curvature are not naturally reproduced, and empirical fixes for this have not been very reliable" [6]. For this reason, the solution of the boundary layer equations using stress-transport models is not addressed in this book.

1.1 CS Model For two-dimensional incompressible flows, the continuity and momentum equations are given by [2] au 8v ( 1.1.1 )

x}8y=0

2u

u8x+vauy

lax+vay2 A

_a y

With Bernoulli's equation, the above momentum equation can be written as

1.1 CS Model

3

8u

8u

_

due

8

82u

+ u8x + v8y _ ue dx v8y2

8y ( u

Y)

(1 . 1 3) .

The boundary conditions for Eqs. (1.1.1) and (1.1.3) are

u=0,

y=0,

y=6,

v=0,

u=1Le.

(1.1.4a)

(1.1.4b)

The Reynolds shear stress term in Eq. (1.1.3), that is, -cu'v', requires a closure assumption. In the CS model this is achieved by using Eq. (1.0.1) with defined by separate expressions in the inner and outer regions of the boundary layer. Excluding the low-Reynolds-number and mass-transfer effects and assuming flow over a smooth surface (no roughness effects), the CS eddy viscosity model is given by 11j. Inner region: 0 < y C yc (Em)i = Z2

au

8y l'tr

(1.1.5)

Here the mixing length l is given by

I = rcy [1 - exp (-A)I

(1.1.6a)

where n = 0.40 and A is a damping-length constant, which may be represented by

A=26Lu1 NT

N=(1-11.8p+)1/2 p+=Vuedue, ur dx

(1.1.6b)

Outer region: yc < y < S to

(E'00 = a fo (Ue - u)dYYtrt

(1.1.7)

Here y accounts for the intermittency of the outer region and is represented by

y = 2 11 - erf [Y - a

(1.1.8)

where Y and a are general intermittency parameters, with Y denoting the value of y for which y = 0.5, and a the standard deviation. The dimensionless intermittency parameters Y/6* and a/S* are expressed as functions of H as shown in Fig. 1.1a. The variation of the ratio of boundary-layer thickness 6 to S* with H is shown in Fig. 1.1b. The parameter a is calculated from

a=

0.0168 C9 U au

t?y)m11.5

l

1. Turbulence Models

4

a 6'

1.2

(a)

1.4

1.6

1.8

20

2.2

3

H

1.5

(b)

1.7

1.9

21

23

H

Fig. 1.1. Variation of Y/6', a/6' and 6/5 with H according to the data of Fiedler and Head [7].

Here subscript m denotes the location where turbulent shear is maximum. The parameter Q is given by 6

0=

1 + 2Rt(2 - Rt)

Rt<1.0 (1.1.10)

1+R t

Rt > 10

Rt

Here Rt denotes the ratio of wall shear to maximum Reynolds shear stress [1],

Rt =

Tw

(1.1.11)

(-Q(u'Z'))m In Eqs. (1.1.5) l'tr is an intermittency factor which represents the streamwise region from the onset of transition to turbulent flow. It is defined by the following expression dx ?tr = 1 - exp [_G(x - xtr (1.1.12) 2tr ue

where Xtr is the location of the start of the transition and the factor C is given empirically by 3

C = C2 yz Rxer.34

(1.1.13a)

with Ry,r denoting Reynolds number, RXlr = (uex/v)tr, and C a constant with a recommended value of 60 for high Reynolds flows. For lower Reynolds number, C is given by [1] C2 = 213(log R5 - 4.7323) (1.1.13b)

1.2 k-e Model

5

1.2 k-e Model The k-E model due to Jones and Launder (8) is also based on the eddy viscosity concept, with e,,, given by Eq. (1.0.4). Here cµ denotes a constant and k and E are obtained from differential equations which represent the transport of turbulence kinetic energy, k, and its rate of dissipation, e. They are given by [1, 4]

Dk

_ a

Dt

[ (v

axk

_ a Dt - axk

DE

aui auj ) aua ate; + axi axj ,C_ (8-., aujxy aua Em. ((xj T ( J iix

E,,, \ ak 1 + ak J SXk J + aE

E

+ a£

a2k ] + Ce,

- Ce2

E2 (1.2.2)

For boundary-layer flows at high Reynolds number, Eqs. (1.2.1) and (1.2.2) become

ak ae

ae

u ax

+v ay

-A)

ak _ a

uax+v ay a

(E,,,, aE'

ay

au 2

ay\aksy+Em\ay/ E

(au 2 ae sy/ +c"k E m \ay )

uE

(1.2.3)

e2

-cEZ k

(1 . 2 . 4)

The parameters cµ, Ce Ce21 ak and a£ are given by cµ = 0.09,

cey = 1.44,

ce2 = 1.92,

ak = 1.0,

ae = 1.3

(1.2.5)

These equations apply only to free shear flows. For wall boundary-layer flows, they require modifications to account for the presence of the wall. Without wall functions, it is necessary to replace the true boundary conditions at y = 0 by new "boundary conditions" defined at some distance yo outside the viscous sublayer to avoid integrating the equations through the region of large y gradients near the surface. Usually yo is taken to be Y

YO = UT

YO +,

yo being a constant taken as about 50 for smooth surfaces. For the velocity field, the boundary conditions at y = yo use the law of the wall and require that

uo=ur(11nyol+cl Y !£

(1.2.6a)

//

vo = uoyo - -dur ur

(1.2.6b)

dx

Here c is a constant about 5 to 5.2. Equation (1.2.6b) results from integrating the continuity equation with u given by the law of the wall

u+ = U ur = 01(y+)

(1.2.7)

1. Turbulence Models

6

We also use relations for the changes in shear stress between y = 0 and y = yo in order to calculate uT from

ur2 =

TO

- ayo

(1.2.8a)

10

where To is calculated from

To = (v+en)

aul

y Jvo

with a semi-empirically given by z

a=0.3 The friction velocity ur (° T = Tw +

o-uedxe

(1.2.8b)

T;/e) can also be calculated from

dxy

+v

I- I

dx Joy*

U-r

dy

(1.2.8c)

with u/ur determined from Thompson's velocity profile given by

u+ _

y+ < 4

y+,

(1.2.9)

c1+c2lny++c3(lny+)2+c4(lny+)3, 4
where cl = 1.0828, c2 = -0.414, c3 = 2.2661, c4 = -0.324. For y+ > 50, we can use the logarithmic velocity profile, [1, 21. u.H

= 1 In y+ + c K

(1.2.10)

where r, = 0.41 and c = 5.0. There are several ways to specify the "wall" boundary conditions for k and e. A common one for k makes use of the relation between shear stress r and k. It is given by Bradshaw [1]. To

ko =

Y = yo,

a1

(1.2.11)

where a1 = 0.30. With ro defined by Eq. (1.0.1) and em, by Eq. (1.0.4), Eq. (1.2.11) becomes a1

_

-

ko cR

so

(8u1

(1.2.12)

ay J o

The boundary condition fore can be obtained by equating the eddy viscosity given by the CS model, (E,,,,)CS, to the eddy viscosity definition used in the k-e model, Eq. (1.0.4) which, with low Reynolds number correction, can be written as

(em)CS = cµfµ

kz

(1.2.13)

1.2 k-e Model

7

Here fm is a specified function discussed later in this section. Thus,

J=yo

(1.2.14)

where I is given by Eq. (1.1.6a). The edge boundary conditions for the k-e model equations, aside from the edge boundary condition for the momentum equation,

u-'ue(x)

y-'6,

(1.2.15)

are

e- ee

k 4ke,

y -'b,

(1.2.16)

To avoid numerical problems, ke and ee should not be zero. In addition, ke and E. cannot be prescribed arbitrarily because their development is governed by the transport equations (1.2.3) and (1.2.4) written at the boundary-layer edge, ue ue

dke

de,

= -e

(1.2.17a) 2

= -eel e

(1.2.17b)

e

The above equations can be integrated with respect to x with initial conditions corresponding to kep and Eeo at xo. The solution provides the evolutions of k(x) and e(x) as boundary conditions for the k- and e-equations.

Low-Reynolds-Number Effects To account for the presence of the wall, it is necessary to include low-Reynoldsnumber effects into the k-E model. Without such modifications, this model fails to predict the sharp peak in turbulence kinetic energy close to the surface for pipe and channel flow as well as fails to predict a realistic value of the additive constant c in the law of the wall.

There are several approaches that can be used to model Eqs. (1.2.3) and (1.2.4) near the wall region. For an excellent review of these models, see Wilcox [4] and Patel et al. [9). Before we discuss the models considered here, it is useful to write the k-E model equations in the following general form, Ok

Ok

u Fx I v aY

_ 8 FY

8E

_ a

L

ay

f

FY

C

e

-j where D, E and F as well as ak, o CAE-

(5)2

8k1 v+ e,n) ak

l

lCv+ ae)

CAE

l

E'

en

-+(s D) + F

Ou 2

j-) -cc2f2

(1 . 2 . 18)

£2 ,

+E (1.2.19)

, c61, ce2, f1, f2 are model dependent and

1. Turbulence Models

8

e = s - D

(1.2.20)

Of the several models to account for low-Reynolds-number effects, we consider the models of Chien [10] and Hwang and Lin [11] in the computer program given in Chapter 4. Other models discussed in (1, 41 can also be incorporated into this computer program. The wall boundary conditions for Eqs. (1.2.18) and (1.2.19) are

y=0,

k = 0,

9=0

(1.2.21)

According to Chien's model, the parameters D, E, F, fµ, fl, f2, 0k, ae, cE, and Cr2 are given by

D=2v7 k,

E=-2v(y)exp(-'y+), F=0 f2 = 1 - 0.22 exp (__q)

fl=1.0,

fv = 1 - exp(-0.0115y+),

Qk = 1.0, £ = 1.3,

cc, = 1.35,

3s(1.2.22a)

ce2 = 1.8

where

RT =

k2

(1.2.22b)

ev

According to Hwang and Lin's model, the parameters in Eqs. (1.2.18) and (1.2.19) are given by //

D=2v1 F

"y) fi=1.0,

`2 I

,

f = 1 - exp(-0.01yA - 0.008y3), 1

f2=1.0,

vE = 1.3 - exp[-(ya/10)], ya

ak = 1.4 - 1.1 exp[-(ya/10)], ce, = 1.44,

(1.2.23)

c£2 = 1.92,

y vk/e

Another approach to include the low-Reynolds-number effects in the k-F model is to employ a simpler model near the wall (a mixing-length model [12]

or a one equation model [13] which is valid only near the wall region) and the full two-equation model in the outer region of the boundary layer; the two solutions are matched at a certain point in the boundary layer. This approach is sometimes referred to as the two-layer method or zonal approach. We shall use the latter name as we discuss the solution of the k-E models with the CS model near the wall region.

1.3 k-w Model

9

1.3 k-w Model Like the k-E model discussed in the previous section, the k-w model is also popular and widely used. Over the years, this model has gone over many changes and improvements as described in (4]. The most recent model is due to Wilcox [4] and is given by the following defining equations. defined by With k

Em=w

(1.3.1)

the turbulence kinetic energy and specific dissipation rate equations are 0

a-

Dt Dw

L\

_ a Kv

Dt

axk

1a

vk

+

k ] + Rik axk -

e) aw 1 aw J axk J +

awRaui fk Oxk -/3w 2

(1.3.2)

(1.3.3)

where Rik is given by

aut

Rik = Em

auk 1

(1.3.4)

\ axk + ax,

and

13, 25,

ao =

/3of1,

/1` _ /3o,fa,

1+70X, fQ = 1 +

125

80X;,,,

100

fp

X, _

Q, = 2

1707 SkiI

(fw)3

Xk < 0

1,

9

ak = 2,

1 + 680x2

1 ak Ow XA:

3 axj axj

1 + 400X2' Xk > 0

(1.3.5a) (1.3.5b)

(1.3.5c)

The tensors Qij and Ski appearing in Eq. (1.3.5b) are the mean rotation and mean-strain-rate tensors, respectively, defined by

_ 92'i

1

2

8ui _ auk exi }

(ax3

'

Ski =

1

2

auk + aut axk }

(axi

(1.3.6)

The parameter x, is zero for two-dimensional flows. The dependence of 8 .on Xu, has a significant effect for round and radial jets [4]. This model takes the length scale in the eddy viscosity as (1.3.7a)

and calculates dissipation e from e _ /3"wk

(1.3.7b)

1. Turbulence Models

10

Wilcox's model equations have the advantage over the k-e model that they can be integrated through the viscous sublayer, without using damping functions. At the wall the turbulent kinetic energy k is equal to zero. The specific dissipation rate can be specified in two different ways. One possibility is to force w to fulfill the solution of Eq. (1.3.3) as the wall is approached [4]:

w-Y2

as y->0

(1.3.8)

The other (14] is to specify a value for w at the wall which is larger than w,,, > 100R,y

where Rv, is the mean vorticity at the wall. Menter [14] applied the condition of Eq. (1.3.8) for the first five grid points away from the wall (these points were always below yT = 5). He repeated some of his computations with ww = 1000Ru and obtained essentially the same results. He points out that the second condition is much easier to implement and does not involve the normal distance from the wall. The choice of freestream values for boundary-layer flows is

AL

,

(Em)cc < 10-2(Ernnax,

kx = (Em)oowoo

(1.3.9)

where L is the approximate length of the computational domain and u. is the proportionality. A = 10 has been recomcharacteristic velocity. The factor of mended (14].

For boundary-layer flows, Eq. (1.3.2) reduces to Eq. (1.2.18) with D = 0,

0.09;A. k

(1.3.10)

The specific dissipation rate equation. Eq. (1.3.3) becomes C9,

aw

ux+V

c?

Em

(Jia.'l

(thi )1l2

-aow2

(1.3.11)

1.4 SST Model The SST model of Menter [14] combines several desirable elements of existing two-equation models. The two major features of this model are a zonal weighting

of model coefficients and a limitation on the growth of the eddy viscosity in rapidly strained flows. The zonal modeling uses the 1993 version of Wilcox's k-w model near solid walls and Launder and Sharma's k-s model [1,15] near boundary layer edges and in free shear layers. This switching is achieved with a blending function of the model coefficients. The shear stress transport (SST) modeling also modifies the eddy viscosity by forcing the turbulent shear stress

1.4 SST Model

11

to be bounded by a constant times the local turbulent kinetic energy. This modification, which is similar to the basic idea behind the Johnson-King model [16], improves the prediction of flows with strong adverse pressure gradients and separation. Iii order to blend the k-w model and the k-e model, the latter is transformed into a k- w formulation. The differences between this formulation and the original k-w model of 1993 are that an additional cross-diffusion term appears in the w-equation and that the modeling constants are different. Some of the parameters appearing in the k-w model are multiplied by a function F1 and some of the parameters in the transformed k-e model by a function (1- F1) and the corresponding equations of each model are added together. The function F1 is designed to be a value of one in the near wall region (activating the original model) and zero far from the wall. The blending takes place in the wake region of the boundary layer. The SST model also modifies the turbulent eddy viscosity function to improve the prediction of separated flows. Two-equation models generally under-

predict the retardation and separation of the boundary layer due to adverse pressure gradients. This is a serious deficiency, leading to an underestimation of the effects of viscous-inviscid interaction which generally results in too optimistic performance estimates for aerodynamic bodies. The reason for this deficiency is that two-equation models do not account for the important effects of transport of the turbulent stresses. The Johnson-King model [16] has demonstrated that significantly improved results can be obtained with algebraic models by modeling the transport of the shear stress as being proportional to that of the

turbulent kinetic energy. A similar effect is achieved in the SST model by a modification in the formulation of the eddy viscosity using a blending function F2 in boundary layer flows [4,14]. In the SST model, the eddy viscosity expression. Eq. (1.3.1), is modified, em

alk max( a 1w, QF2)

(1.4.1)

where al = 0.31. In turbulent boundary layers, the maximum value of the eddy viscosity is limited by forcing the turbulent shear stress to be bounded by the turbulent kinetic energy times al, see Eq. (1.2.11). This effect is achieved with an auxiliary function F2 and the absolute value of the vorticity D. The function F2 is defined as a function of wall distance y as F2 = tanh(arg2)

(1.4.2a)

arg2 = Max (2 'Vlk ; 600v1 0-09W Y yew J

(1 . 4. 2b)

where

1. Turbulence Models

12

The two transport equations of the model for compressible flows are defined below with a blending function F1 for the model coefficients of the original w and e model equations. DAk

Dt

8k

_ 8

luui

(A+vkpE)axkJ

axk

Dgw

Dt

_ 8 r

+ awAEm)

8xk

+ 2(1 - FOP-7-2

1

8w 1

Awk

(1.4.3)

8i

-y

axti J + E Rik axk

8k 8w (1.4.4)

W OXk OXk

where

-

+ auk 2 8uj bik) 2 gk6ik (1.4.5) axi 3 axj 3 The last term in Eq. (1.4.4) represents the cross-diffusion (CD) term that apC7ui

Rik = QED,,

-

axk

pears in the transformed w-equation from the original E-equation. The function

F1 is designed to blend the model coefficients of the original k-w model in boundary layer zones with the transformed k-E model in free shear layer and freestream zones. This function takes the value of one on no-slip surfaces and near one over a larger portion of the boundary layer, and goes to zero at the boundary layer edge. This auxiliary blending function F1 is defined as F1 = tanh(arg1

arg1 = min max 1

500v

0.09w y

(1.4.6)

4Qaw2k

JZw } ' CD kwy2

(1.4.7)

where CDk,,, is the positive portion of the cross-diffusion term of Eq. (1.4.4): 1

CDku, =max 2Qaw2 W

8k 8wxk' 8xk 8

10_20 1

(1.4.8)

The constants of the SST model are ,0' = 0.09,

n = 0.41

(1.4.9)

The model coefficients,3, -y, ak and au , denoted with the symbol 0, are defined by blending the coefficients of the original k-w model, denoted as 01, with those of the transformed k-E model, denoted as 020 = F101 + (1 - FI)02 where

0= fake aw,A-Y} with the coefficients of the original models defined as inner model coefficients

(1.4.10)

1.5 SA Model

13

awl = 0.5,

ak1 = 0.85,

/31 = 0.075

2

* -awl K

ry1 =

(1.4.11)

= 0.553

outer model coefficients awe = 0.856,

ak2 = 1.0, rye =

N2

-

/32 = 0.0828 (1.4.12)

aw2 K2

77 = 0.440

The boundary conditions of the SST model equations are the same as those described in the previous section for the k-w model. For incompressible boundary-layer flows, Eq. (1.4.3) reduces to the kinetic energy equation given by Eqs. (1.2.18) and (1.3.2). Equation (1.4.4) is same as Eq. (1.3.11) except that its right-hand side contains the cross diffusion term,

+ 2(1 - F1)aw2-

k aw

(1.4.13)

y ay Two-equation models are not entirely reliable in complex flows. The "v2f" model of Durbin [171 gives generally better results, at the expense of using one more transport equation, nominally for the -ev72 Reynolds stress, and an elliptic relaxation equation to represent the effect of pressure fluctuations.

1.5 SA Model Unlike the Cebeci-Smith model which uses algebraic expressions for eddy viscosity, this model uses a semi-empirical transport equation for eddy viscosity. Its defining equations are as follows. Cm = vtfui

Dvt

tt

Dt - cb1 l1 -

c

1

(1.5.1)

2

(cwtfw - Z ft2) (0d ) +

l a 1

av

axk [(V +vt)

cb2 ai;t arlt

(1. 5.2)

a axk axk Here cb1 = 0.1355, cb,

cw 1 =

fvl=X

2

+

(1

c62 = 0.622,

acb2)

,

cv1 = 7.1.

c,,,2 = 0.3,

3 +c3l'

fv2=1-1+Xfv,,

a=

ct g = 2,

2

(1.5.3a)

,c = 0.41

r 6+ 6 + fw=g 9

(1.5.3b)

1/6

(1.5.3c) 3

1. Turbulence Models

14

vt

x= v,

g=r+cu,2(r6-r), L

S+K2afV2, z

r =

(1.5.3d)

2i2 i2

S

ct3 = 1.1,

ft2 = et3e-"4X ,

Vt

Sfc2d2 (1.5.3e)

ct4 = 2

(1.5.3f)

where d is the distance to the closest wall and S is the magnitude of the vorticity,

The wall boundary condition is Pt = 0. In the freestream and as initial condition 0 is best, and values below io are acceptable [3]. For boundary-layer flows, Eq. (1.5.2) can be written as 8vt U-

8vt

8

1

+ V-- =ebi(1-ft2)SVt+1

-(

cf-

where

S=

8vt

rapt

)l2

L(v+vt)8+ jc / ll

t

l ()2

ft2) 8u

vt

8y

+ k2d2 fv2

(1.5.4)

(1.5.5)

References [1] [2] [3] [4]

[5] [6] [7]

(8]

Cebeci, T.: Analysis of Turbulent Flows, Elsevier, 2003. Cebeci, T. and Cousteix, J.: Modeling and Computation of Boundary-Layer Flows, Horizons Publishing, Long Beach, CA and Springer, Heidelberg, Germany 1998. Spalart, P. R. and Allmaras, S. R.: A One-Equation Turbulence Model for Aerodynamics Flows. AIAA Paper 92-0439, 1992. Wilcox, D. C.: Trbulence Modeling for CFD. DCW Industries, Inc., 5354 Palm Drive, La Canada, Calif., 1998. Launder, B. E., Reece, G. J. and Rodi. W.: Progress in the Development of a Reynolds Stress Turbulence Closure. J. Fluid Mech. 20, 3, 1975. Bradshaw, P.: The Best Turbulence Models for Engineers. Modeling of Complex Flows (M. D. Sales, J. N. Hefner and L. Sakell, eds.) Kluwer, Dordrecht, 1999. Fiedler, H. and Head, M. R., "Intermittency Measurements in the Turbulent Boundary Layer," J. Fluid Mech. 25, 719-735, 1986.

Jones, W. P. and Launders, B. E.: The Predicition of Laminarization with a TwoEquation Model of Turbulence. Int. J. Heat and Mass Transfer 15, 301-314, 1972.

Patel, V. C., Rodi, W., and Scheuerer. G.: Turbulence Models for Near-Wall and Low Reynolds Number Flows: A Review. :1IAA J. 23,1308-1319,1985. [10] Chien, K. Y.: Predictions of Channel and Boundary-Layers Flows with a Low-ReynoldsNumber Turbulence Model. AIAA J. 20, 33-38, 1982. [11] Hwang, C. B. and Lin, C. A.: Improved-Low-Reynolds Number k-s Model Based on Direct Numerical Simulation Data. AIAA J. 36, 38-43, 1998. (12] Arnal, D., Cousteix, J., and Michel, R.: Couche limite se developpant avec gradient de pression positif dans un ncoulement turbulent. La Rech. Aerosp. 1976-1, 1976. (9)

References

15

[13] Norris, L. H. and Reynolds, W. C.: Turbulent Channel Flow with a Moving Wavy Boundary. Report FM-20, Department of Mechanical Engineering, Stanford University, Stanford, California, 1975. [14] Mentor, F. R.: Two-Equation Eddy Viscosity Turbulence Models for Engineering Applications. AIAA J. 32, 1299 -1310, 1994. [15] Launder, B. E. and Sharma, B. I.: Application of the Energy Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc. Letters in Heat and Mass Transfer 1, 131-138, 1974. [16] Johnson, D. A. and King, L. S.: A Mathematical Simple Turbulence Closure Model for Attached and Separated Turbulent Boundary Layers. AIAA J. 23, 1684-1692, 1985. (17] Durbin, P. A. and Petterson Reif, B. A.: Statistical Theory and Modeling for Turbulent Flows. John Wiley and Sons, New York, 2001.

Solution of the CS and k-E Model Equations

2.0 Introduction In this chapter we discuss the numerical solution of the boundary-layer equations

using the CS and k-e models, discussed in Section 1.1 and 1.2 respectively. In Section 2.1 these equations are expressed in transformed variables, which stretch the coordinate normal to the flow and allow large steps to be taken in the x-direction. Section 2.2 discusses the solution procedure for the k-e model equations with and without wall functions as well as with a zonal method. Numerical solution of the k-e model equations is addressed in Section 2.3 for a zonal method and in Section 2.4 for model equations with and without wall functions.

2.1 Transformed Variables We use the Falkner-Skan transformation discussed, for example, in [1]. With the similarity variable defined by 7Le

77=

vxy

(2.1.1a)

and the dimensionless stream function f (x, n) by

0(x, J) = uetixf (x, 77),

(2.1.1b)

the continuity and momentum equations, Eqs. (1.1.1) and (1.1.2) and the k-e model equations, Eqs. (1.2.18) and (1.2.19), with k and a defined by

u

,

e=ue,

(2.1.2)

2. Solution of the CS and k-e Model Equations

18

and a prime denoting differentiation with respect to 77, can be written as

- f/1afl

(bf")' +miff"+m(1-(f')21 =x(f'

(2.1.3)

(b2k')'+P-Q+F=2muk-mlfk'+x(u

(2.1.4)

of

(b3,-')' + P1 - Qi I E = x uax - £'ax) + (3m - 1)uE - mlCIf

(2.1.5)

where the tilde has been dropped from the equations and

b2=1+ m

b=1+e,+,t,

b3=1+ '-n

ak

aE

(2.1.6)

xdue +Em _ 7IL=--, m1= m+1 Bm 2 ue dx v,

In Eq. (2.1.4) (b2k')' denotes the diffusion term, P and Q defined by (f")2,

P=e

Q=e+D

(2.1.7)

denote the production and dissipation terms, respectively. The right-hand side of Eq. (2.1.4) represents the convection term. In Eq. (2.1.5) (b3E')' denotes the diffusion term, P1 and Q1, defined by

P1 = cE'flccff,v2k

(2.1.8a)

Q1 = C62f2e2/k

(2.1.8b)

denote the generation and destruction terms, respectively. The right hand side of Eq. (2.1.5) represents the convection term. The wall boundary conditions depend on whether the above equations are being solved for high Reynolds flows (without wall functions), for the zonal method or for flows at low Reynolds numbers (with wall functions). For high Reynolds numbers the four boundary conditions at y = y0 correspond to Eqs. (1.2.6) (1.2.12) and (1.2.14). In terms of transformed variables, Eqs. (1.2.6) can be written as 1

A = wo I In (

xwo17o) + cJ

ax + ml f0 = fort where WO

=uT,

ue

Rx=uex,

v

(ml

x dwo l w0 dx J

C=5.2,

, =0.41

(2.1.9a) (2.1.9b) (2.1.10)

In terms of dimensionless and transformed variables, Eq. (1.2.12) can be written as

19

2.2 Solution Procedure

fit

al =c,,k

(2.1.11)

all evaluated at 77 = ?70.

The fourth boundary condition, Eq. (1.2.14) becomes K2.q2fo

Rxwoi7o/26)12

{1 - exp(-

J}

2

= c Rx O CO

(2.1.12)

For low Reynolds number flows, the wall boundary conditions are given at 71 = 0, that is (2.1.13a) f = 0 (no mass transfer), f' = 0

k=0, E=0

(2.1.13b)

In the zonal method where we use, for example the CS model near the wall region 0 < y < yo, the wall boundary conditions for f and f at 77 = 0 are given by Eq. (2.1.13a) and those at 77 = rlo by Eqs. (2.1.11) and (2.1.12). In all cases, the "edge" boundary conditions at 77 = 77e are given by 77=7le,

u=1.0,

(2.1.14a)

z

xax + e + 2mk = 0,

x ae + ce2 f2

l2

+ (3m - 1)E = 0

(2.1.14b)

2.2 Solution Procedure The solution procedure of the k-E model equations can be obtained with and without wall functions. It can also be obtained with a zonal method. In this case the boundary layer equations axe solved in two regions with each region employing different turbulence models. In effect this approach may be regarded as the use of the k-e model with wall functions. In this book we use Keller's Box method to solve the boundary-layer equations. This is a second order two-point finite-difference method described in detail in several references, see for example [1]. In this method the governing equations are first expressed as a first order system by introducing new functions to represent the derivatives of f, k and e with respect to rl. The first-order equations are approximated on an arbitrary net, Fig. 2.1, with "centered-difference" derivatives and averages at the midpoints of the net rectangle difference equations. The resulting system of equations which is implicit and nonlinear is linearized with Newton's method and solved by the block-elimination discussed in Section 2.4 and also in [1]. With the introduction of new variables

f' = u

(2.2.1a)

u' =V

(2.2.1b)

2. Solution of the CS and k-e Model Equations

20

`n-1

Xn-112

Xn-1

Xn

X.

Fig. 2.1. Net rectangle for difference approximations

k' = $ E' = q,

(2.2.1c)

(2.2.1d)

the k-e model equations given by Eqs. (2.1.3) to (2.1.5) can be reduced to a system of seven first-order equations given by Eqs. (2.2.1) and by

(bl v)' + ml f v + m(1 - u2) =X Cu

au

- v Of

J

(2.2.2)

/ (b2s)'+P-Q+F-2muk+mlfs= (uOx-sax J

(2.2.3)

8x

8x )

(b34)'+Pi-Qi+E+mlfq-(3m-1)ue=x(uax-qOf

I

(2.2.4)

As discussed above, the k-e model equations without wall functions use "wall" boundary conditions specified at some distance 7710 outside the viscous

sublayers. In this case the boundary conditions on f' and f (or u and f) are represented by Eqs. (2.1.9), that is

1 In (

up = w0

1l

(2.2.5a)

Rkwo'+1o) + cJ

L

8fo

x8x + ml fo =

x dwol uni IM + w dx 0

(2 . 2 . 5a)

l

and those for k and a by Eqs. (2.1.11) and (2.1.12), that is (2.2.6)

1`

eo 2

(em)CS = ie2rlo {1 - exp(-

R,r woilo/26) }2 vo = cU Rx Lo

(2.2.7)

0

although in the latter case, there are other choices. In either case, the friction e) appearing in the it and f equations is unknown and must velocity, uT (

2.2 Solution Procedure

21

be determined as part of the solution. One approach is to assume uT, (say from the initial profiles at the previous x-station), and solve the governing equations subject to the "wall" and edge boundary conditions. From the solution determine To at yo, (2.2.8)

To =

ay )o and compute u . from Eq. (1.2.8). If the calculated value of uT does not agree with the estimated value within a specified tolerance parameter 61,

uT+1 - url < 61

(2.2.9)

then a new solution is obtained with the updated values of uT and To. This procedure is repeated until convergence. This iterative procedure can be replaced with a more efficient one by treating uT as an unknown. Since uT is a function of x only, we can write

u,. = 0

(2.2.10)

thus increasing the number of first-order equations from seven to eight. Although we now solve eight first-order equations rather than seven, this procedure allows the solutions to converge faster, especially for flows with strong adverse pressure gradient.

In the solution procedure described here the numerical method is formulated for eight unknowns, not only for the k-c model equations with the zonal approach but also for the k-e model equations with and without wall functions. This choice does not increase the complexity of the solution procedure, but paves the way to solve the k-a model equations or others in an inverse mode if the solution procedure is to be extended to flows with separation (see Section 6.3).

In Keller's Box method, the x-wise derivatives are represented by central differences. Experience with the Box method has shown that when profiles are used to start the turbulent flow calculations, the solutions at the subsequent xlocations oscillate. A common cure to this problem is to compute the first two x-stations equally spaced and take an average of the solutions at the midpoint of xo and xl, say x,,, and xl and x2, say xe. Then another average of the solutions is taken at x.,,,, and xe defining a new solution at xl. When new calculations

begin at x2 with averaged profiles at x = xj, the solutions at x > X2 do not exhibit oscillations.

While this cure is relatively easy to incorporate into a computer program and in most cases provides stable solutions in adverse pressure gradient flows sometimes the solutions may break down due to oscillations. On the other hand, the author and his colleagues observed that if one uses backward difference approximations for the x-derivatives in the boundary-layer equations, rather than central differences as used in the Box method, the solutions do not oscillate

2. Solution of the CS and k-E Model Equations

22

and are more stable. For this reason, when the Box method is used for turbulent flow calculations with initial profiles, we will represent the x-derivatives with backward finite-difference approximations.

2.3 Solution of the Ic-e Equations with a Zonal Method In this method, the boundary-layer is divided into two zones. The inner zone is identified by y < yo, yo = (you, /v) - 100, where the continuity and momentum

equations, Eqs. (1.1.1) and (1.1.3), are solved subject to the wall boundary conditions given by Eq. (1. 1.4a), with eddy viscosity e,,,, given by the inner region

of the CS model. In the outer zone, y > yo, the equations for continuity Eq. (1.1.1), momentum Eq. (1.1.3), turbulence kinetic energy Eq. (1.2.3) and rate of dissipation Eq. (1.2.4) are solved subject to the inner boundary conditions given by Eqs. (1.2.11) and (1.2.14) and the edge boundary conditions given by Eqs. (1.2.15), (1.2.16) and (1.2.17), with eddy viscosity E,n, computed from Eq. (1.0.4).

2.3.1 Inner Region The numerical solution of the k-e model equations with the zonal method requires that in the inner region Eq. (2.1.3) is solved subject to the true wall boundary conditions f = 0, u = 0. Since, however, the solution procedure is being formulated for the general case which includes the solution of the k-e model equations without wall functions, it is necessary to specify a boundary condition for u,.. This can be done as described below.

From the definition of uT (= ue 2 ), we can write uT uc

w=

or in transformed variables,

w=

f,,w

cf

2

(2.3.1a)

2 3 1b)

The boundary condition for w is

too = i4 Rz

(2.3.2)

Next the eight first-order equations can be written by letting u' = v, k' = 0, s' = 0, e' = 0, w' = 0, q' = 0, f' = u and the momentum equation (2.2.2). For j = 0, with the first three equations corresponding to boundary conditions, the equations for the inner region are ordered as

2.3 Solution of the k-e Equations with a Zonal Method

23

fo = 0

(2.3.3a)

ue = 0

(2.3.3b)

WO =

(2.3.3c)

1/4

U/ = v

(2.3.3d)

k' = 0 s' = 0

(2.3.3e)

e' = 0 q' = 0

(2.3.3g)

(2.3.3f) (2.3.3h)

With finite-difference approximations and linearization, they become bfo = (r1)o = 0

(2.3.4a)

buo = (r2)o = 0

(2.3.4b)

bvo - 2 R;wobwo =(1'3)0 = h?

by +bvj_

r-

vo

(2.3.4c)

(2.3.4d

bkj - bkj_1 = (rs)j = 0

(2.3.4e)

68j - 63j-1 = (7-6)j = 0

(2.3.4f)

(2.3.4g) Ssj - bej_1 = (r7)j = 0 bqj - bgj_1 = (r8)j = 0 (2.3.4h) For 1 < j < j8i the order of the equations is the same as those above, except that the first three equations are replaced by

w' = 0

(2.3.5a)

f'=u

(2.3.5b)

momentum Eq. (2.2.2)

(2.3.5c)

which, in linearized form can be written as

5wj - 8wj_1 = (r1)j = 0

bfj - bfj-1 -

J2

(buj + buj_1) = (r2)j = fj-1 - fj + hjuj-1/2

(2.3.6a) (2.3.6b)

(s1)jbfj + (s2)jbfj_I + (83)j6uj + (84)jbuj_1 (2.3.6c)

+ (85)jbvj + (86)jbvj_l = (r3)j

The finite-difference procedure for Eq. (2.2.2) is identical to the procedure used in the Box method for the momentum equation described in [1]. The only

2. Solution of the CS and k-e Model Equations

24

difference occurs in the solution of Eq. (2.2.2) where we use three-point or twopoint backward finite-difference formulas for the x-wise derivatives rather than central differences as was done in [1]. For this purpose, for any variable V, the is defined by derivative of n

(

= Al

57x

Vix-2

+ A2Vn-1 + A3Vn

(2.3.7)

where for first-order

Al=0, A2=- xn - xn-1 1

and second-order

A = 1

A2

,

A3=

1

xn - xn-1

(xn - xn-1) (xn_2 - xn-1)(xn-2 - xn) (xn - Xn-2) xn_1 - xn-2)(xn-1 - xn) 2xn - xn_1 - Xn-2

(2.3.8)

(2.3.9)

(/

A3

_

(xn - xn-2)(xn - xn-1)

Representing the x-derivatives in Eq. (2.2.2) with either two-point or three-point backward difference approximations at x = xn and using central differences in the 77-direction, we can write Eq. (2.2.2) as h; 1[(bv)a

- (bv)j_1] +ml(fv)j 112+mn[1 - (u2)j_1,21 xn [(V f n (v8f)n n xn a(u2) + ax ax ax ;_1 Ij-1/2 2

a

L

(2.3.10)

Linearizing we get

h;1(bjndvj - bj 1Svj_1) n

+ 2(f?bv;+v?bf;+f? 1bv;_1+v? 16fj_1) - mn(ujbuj + uj_lbuj_1) n 4

(a

a (5n2

n

buy

(2.3.11)

6u;_1

+ au ax) j-1 / `j - 2 l X Jnbv;+v;a (axbff+(ax}n j f j-1 au

n

+ v?

1

f ()'

ax ;_1

6fj_1 + (r3)j

From Eq. (2.3.7), it follows that

2.3 Solution of the k-e Equations with a Zonal Method

(2)1t

n

2

a (ax) j-1 =

=2A.3uj,

j

a

_

= A3,

ff (ax2--)i

25

fn

2A3uj-1

(2.3.12)

= A3

of

The linearized expression can be written in the form given by Eq. (2.3.6c). The coefficients (sl)j to (s6)j and (r3)2 are given by

(si) j = 2 (?7i,' + xnA3)v?

(2.3.12a)

(82)3 = 2(mi +xTA3)vj 1

(2.3.12b)

(Mn +

(2.3.12c)

(s3)j

(mn +

(s4)j

2

A3) uj-

u

2 A3/

mn

(s5)j = hj bb + 2f j +

x/n

/a f\n I

2

(2.3.12d)

1

x

(2.3.13e)

!

(s6)j=h?lb3 1+21 fj 1+ 2

(2.3.13f)

(r3)j = - [h, 1 [(bv)a - (bv) 1) + mi (fv)7 1/2 + M'11

+ i xn I

a (u2>J Lax j_1/2

- (u2)7 1/21

\n

n

1n

r

tvOxf J+ i v of ex J j

2

n

j-1 (2.3.14)

The linearized finite-difference equations and their boundary conditions, Eqs. (2.3.4) and (2.3.6) are written in matrix-vector form as described in [11

Ab=r

(2.3.15)

where

A=

AO Co

So

r'a

B1 Al C1

b1

r1

Bj

Aj

b=

C3

bi

r=

''j

Bj-1 AJ-1 CJ-1

Bj

Aj

I

I

bjI

I

1'j

(2.3.16)

2. Solution of the CS and k-e Model Equations

26

The eight dimensional vectors 5j and r'j for each value of j are defined by S fj

(ri)j

Suj

(r2)j

Svj

(r3)j

Skj

rj =

I

Ssj

(r4)j

(2.3.17)

(r5)j

Ssj

(r6)j

8qj

(r7)j

Swj

(r8)j

The definitions of 8 x 8 matrices Aj, Bj and Cj in the inner region 0 < j < js are 1

0

0

1

0 0

0

0

1

0 0

0

0

0 -2/w

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

-; (si)j (s3)j Aj =

0 0

0

1

0

0

0 0

0

0 1- 2 0

0

0

0 0

0 -1 - Z 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1

AO =

Ci

0

0 0

0

0<j
0

0

0

0

1

0

0

0

0

0

0

(s5)j

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

(2.3.18a)

0

0

-1 - 2 0

o

0

-1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0

1 < j < js - 1

(2.3.18b)

(2.3.18c)

27

Method

a

Zonal

with

Equations

k-e the

of

Solution

2.3

-1

0 0 0 0 0 /

(S6)j

(S4)j

(2.3.18d)

1<j<j.,

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

00 0 0

=

0 0000

0 0 0 0 0

000 000 000 000 000 0 0

0 -IZ

0 -1 ($2)j

Bj

Regions

Outer

and

Inner

Between

Interface

2.3.2

as

ordered

now

is

equations

of

(2.3.19a)

=

0 u}

(2.3.19b)

=

u

f'

(2.3.19c)

(2.2.2)

Eq.

momentum

(2.2.7)

Eq.

(2.2.6)

(2.3.19e)

Eq.

b.c.

(2.3.19d)

b.c.

(2.3.19f)

=

v U/

(2.3.19g)

s k'=

(2.3.19h)

=

q

given

Bj with

equations,

linearized

the

from

matrices

Cj

are

jg

=

j at

(2.3.13)

Eq.

by

to

(s6)j

(si)j

by

Eq.

(2.3.18d)

and and

resulting

system

first-order

The

Aj

The

(2.3.20a)

1 0 0 0 0 0 0 0

(s3)j,

D1 D4

---2

-

0

0 0 0 0

0

-1

0

(2.3.20b)

00 0 0 0

2

1-

0

0 0

1-

20

000 0 0 ±

0 0 0 0

-h;

2

-

0 0 0 0

0 0 0

Cj,

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 =

0 0 0 0 0 -1 D3 D6 0 0 0 0 0 0 0 0 0 0 D2 D5 0 -1 0 0 0 0 0 0 0

(S5)j=

0 0 0 0 0 0 0 0 &-' 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0

-

02

0 1

(S1)js

=

Aj,

2. Solution of the CS and k-E Model Equations

28

Here the fourth and fifth rows of Aje follow from the boundary conditions, Eqs. (2.2.6) and (2.2.7), at ?7 = rlo. After the application of Newton's method to these equations, D1 to D6 are given by the following expressions. Dl = Ejs F (Em)CS

D2 = -2 R. c,,kj,

,

D4 = 2R,,ck vjg1

D5 =

D3 = (e )CS

,

D6 = -2Ejs

kjg,

(2.3.21a) (2.3.21b)

The associated (r4)j, and (r,5)j, axe (r4)js =

Rx cpka - (e')CSEjs

(2.3.22a)

(r5)j. = 623. - R.c,.(kjsvjs)2

(2.3.22b)

2.3.3 Outer Region The finite-difference approximations for the outer region defined for js + 1 < j < J are written by using a similar procedure to that described for the inner region equations. The first-order system of equations is ordered similar to those given by Eqs. (2.3.19) except that Eqs. (2.3.19d) and (2.3.19e) are replaced by Eqs. (2.2.3) and (2.2.4). The resulting matrices from the linearized equations,

with the Cj matrix remaining the same as that given by Eq. (2.3.20b) for

j8<j<J-1,are 0

0

0

0

0

0

-1

0

0

0

0

0

0

(,'6)j

(38)j

0

(S12)j

0

0

0

-1 -j (S2)j (34)j

Bj = (a2)j (a4)j ((a6)j (a8)j (a10)j (aR12)j 0

(fl2)j (04)j (,/36)j (08)j

0 1

0

0

0 0

0 0

0

-

(S1)j (S3)j

Aj =

0 0 0

0 0

0 0

0

0

0

0

0

0 0

0 0

0 0

0

0

0

0

0

1

0

0

0

0

0

0

(S5)j

(S7)j

0

(Sll)j

0

0

R0

0

(al)j (a3)j (a5)j (a7)j (a9)j (all)j (01)j (03)j ($5)j (07)j 0 (,811)j

0 0 0

0 R0

(/312)j (N14)j 0

-1 - 2 0 0 -1 - 2 0 0

0

0

0

0

0

0

-1

(N13)j 0 0

0

0

0

- h '+L

js+1<j<J-1

0 (2.3.23b)

2.3 Solution of the k-E Equations with a Zonal Method

0

0

29

0

0

0

0

0

1

0

0

0

0

0

0

0

(sll)J

0

0

AJ _ (al)J (a3)J (a5)J (a7)J (as)J (0111)J

0

0

-z

(s1)J (s3)J (s5)J (s7)J

0

(fl)J (163)J (f5)J (f7)J

0 0

0

0 0 0

1

0 0

0 E1 E3

(2.3.23c)

(011)J (f13)J 0

0 0 0

0

E2 E4

0 0 0

0 0 0

Here (81)j to (s12)j, (al)j, to (a12)j and (/31)j to (914)j given in subsection 3.3.1 correspond to the coefficients of the linearized momentum (2.2.2), kinetic energy of turbulence (2.2.3), and rate of dissipation (2.2.4) equations written in the following forms, respectively,

(Sl)j8fj + (s2)j6fj-1 + (s3)jbuj + (S4)jbuj-1 + (S5)jbvj + (86)j6vj-1 + (s7)jbkj + (S8)jbkj-1 + (s11)j&j + (s12)j&Ej-1 = (r3)j

(2.3.24)

(al)jbfj + (a2)j6fj-1 + (a3)jbuj + (a4)jbuj-1 + (a5)jbvj + (a6)j6vj-1 + (a7)jbkj + (as)jbkj_l + (a9)j6sj + (aio)jbsj-1 + (a11)j&j + (a12)jbej-l = (r4)j

(2.3.25)

(f1)jbfj + (f2)jbfj-1 + (f3)j6uj + (f4)j6uj-1 + (a5)jbvj + (136)jbvj-1 + (,37)jbkj + (13s)jbkj-1 + (f11)jbcj

+ (f12)j6&5-1 + (f13)jbgj + ()314)j5gj-1 = (r5)j

(2.3.26)

The last three rows of the Aj matrix correspond to the edge boundary conditions and follow from the linearized forms of Eq. (2.1.14). They are given by

El = 2m" + x" E2 = 1,

8 8k

/8kl" 8x J '

E3 = -ce2f2

3m" - 1 + x"

a lE4=

8a (F'X J J

+ ce2 f2

where

8k

8x)J - A3,

(2.3.27)

(k2)11,

8s

x}

kn k-'J A3

(2.3.28)

The coefficients (r7)J and (r8)J are given by

(r7)J = -

+ej+27n"kjI X. (ax)J (rs)J = - fx" (8x)J+ce,f2 (k; +(3m" - 1)ej]

(2.3.29a) (2.3.29b)

2. Solution of the CS and k-e Model Equations

30

2.3.4 Block-Elimination Method The linear system expressed in the form of Eq. (2.3.15) can be solved by the block-elimination method discussed by Cebeci and Cousteix [1]. According to this method, the solution procedure consists of two sweeps. In the first part of the so-called forward sweep, we compute Tj, 4j from the recursion formulas given by

40 = AO

(2.3.30a)

1'j4j_1 =Bj j = 1,2,...,J

(2.3.30b)

4j = Aj -1'jCj_1 j = 1, 2, ... , J (2.3.30c) where the Ij matrix has the same structure as Bj. In the second part of the forward sweep, we compute wj from the following relations (2.3.31a)

zvo = ro

ivj = r"j - rjwj_1

1<j<J

(2.3.31b)

In the so-called backward sweep, we compute Sj from the recursion formulas given by

4jbj = wj 415 = wj - CjSj+1 j = J - 1, J - 2, ... , 0

(2.3.32a)

(2.3.32b)

The block elimination method is a general one and can be used to solve any system of first-order equations. The amount of algebra in solving the recursion formulas given by Eqs. (2.3.30) to (2.3.32), however, depends on the order of the matrices Aj, Bj, Cj. When it is small, the matrices 1'j, 4j and the vector wj can be obtained by relatively simple expressions, as discussed in [1]. However, this procedure, though very efficient, becomes increasingly tedious as the order of matrices increases and requires the use of an algorithm that reduces the algebra internally. A general algorithm, called the "matrix solver", discussed by Cebeci and Cousteix [1] and in Section 3.4 can be used for this purpose. In addition, since the zonal method requires that the linearized inner boundary conditions resulting from Eqs. (2.2.6) and (2.2.7) be satisfied, as well as the usual boundary conditions at the surface and the boundary-layer edge. Subsection 3.3.5 presents an algorithm utilizing the "matrix solver" and called KESOLV for this purpose. It employs the block-elimination method and follows

the structure of the solution procedure used in the zonal method, as well as the procedure used in the solution of the k-E model equations with and without wall functions discussed in the following section.

2.4 Solution of the k-e Model Equations with and Without Wall Functions

31

2.4 Solution of the k-e Model Equations with and Without Wall Functions The solution of the k-e model equations with and without wall functions is similar to the solution of the k-E model equations with the zonal method. Their solution in either case can be accomplished with minor changes to the solution algorithm described in the previous section. In both cases changes are made to the Ana matrix, Eq. (2.3.20a), by modifying or redefining the elements of the first five rows which in this case correspond to the boundary conditions at n = r!o or q = 0. In either case, for j = 0, after the five boundary conditions are specified, the next three equations correspond to those given by Eqs. (2.3.19f) to (2.3.19h). For j > 1, the ordering of the first-order equations is identical to that used for the outer region, that is, the equations are ordered according to those given by Eqs. (2.3.19) except that Eqs. (2.3.19d) and (2.3.19e) are replaced by Eqs. (2.2.3) and (2.2.4), respectively. In addition of course, the coefficients of the equations for the momentum, kinetic energy and rate of dissipation are different.

2.4.1 Solution of the k-E Model Equations Without Wall Functions The transformed k-e model equations without wall functions for high Reynolds number flows are still given by Eqs. (2.2.3) and (2.2.4) provided we set

1'=eMV2, Q=E, F=0 E2

P1=ce,ficµv2k,

(2.4.1)

E=0, fl=f2=1.0.

There are five "wall" boundary conditions; four are given by Eqs. (2.2.5), (2.2.6) and (2.2.7) for 71 = 710. After linearization they can be expressed in the form 6up + a85wo = (rl )o

(2.4.2)

131bfo + 02buo + 1386wo = (r2)0

(2.4.3)

,'3bvo + y46ko +'y66e = (r3)0

(2.4.4)

036vo + 646ko + 9e6eo = (r4)o

(2.4.5)

where as

_- -L

Q2 =-

ln(,/R__xworlo) + c]

+a 1- w

r n.

T 70

L

13 =c,

1

\

ko,

n-1 o

w

xn

01 = a +m1,

kn

(2.4.6a)

w0"`-1

A.

14 =cW R.vo,

- --u oqoa(wOD2 76 =- cA:

(2.4.6b)

(2.4.6c)

2. Solution of the CS and k-e Model Equations

32

83=-cr,EO,

94=2cf, Ryko,

es= -cuvo,

77

damping)2 (2.4.6d)

1

(rl)o = w0 - ln( Ry w0i70) + c - u0

(2.4.7a)

n-1

(r2)o = u0770 m.1 + a

1 -win

- a(fo - fo -1) - m1fo

(2.4.7b)

0

r3 = cu Eo - c,, Vf kovo . T4 =

ctL

(2.4.7c)

(2.4.7d)

Rx ko

If 710 is sufficiently away from the wall, i.e. yo > 60, then the damping term, such as the one used in the CS model, is equal to 1.0. The fifth "wall" boundary condition which connects ro at y = yo and r,,, at y = 0, is obtained from Eq. (1.2.8c). With Thompson's and log law velocity profiles, it can be written as r0=rw+a*

yo

r,

d LP dx +yo dx

(2.4.8)

Here a* is given by

a* = 0.5

[c1(lnY)2+c2lnY+ c3 +

(2.4.9)

c+ YO

I

where cl = 5.9488 ,

c2 = 13.4682 ,

+= y

c3 = 13.5718 ,

c4 = -785.20 (2.4.10)

X,00

a ur 71 0 ,

lV

VTI;

In terms of transformed variables, Eq. (2.4.8), after linearization, can be expressed in the form 6351)0 + 646ko + 666E0 + 686wo = (r5)0

(2.4.11)

where 2

63 = c Rx ke CO

,

64 = 2cp

Rx kovo, CO

2 L2

66 = c Rx E2 zo

(2.4.12a)

68 = -2wo{ Rx+a*rlo(a+2mi)}-

-

(8a*) 710{a[(w0-)2

aw o and

- (wo-1)2] + 2m1(w )2}

(2.4.12b)

2.4 Solution of the k-e Model Equations with and Without Wall Functions

33

Rxw2+a`710{a[(wo)2-(w0-1)21+2mi(w0)2}

(r5)o =

kvo 2 - nomi - cw Rx a

(2.4.13)

0

With the five boundary conditions defined, the Aj9 matrix, which is essentially the A0 matrix in this case, becomes 0

0

as

0

0 0

0

01 /32

0

0

0

638

0

0

73

74

0

Its

0

0

0 o

0 o

03

04

0

06

0

0

63

64

0

66

0

68

0

0

0

0

o

o

0

0

0

AO =

0

1

o -1 -L 0

(2.4.14)

0 -1-4 0 0 0 0 0 -1 -. 0 0

2.4.2 Solution of the k-e Model Equations with Wall Functions The solution of the k-e model equations with wall functions is similar to the procedure described for the case without wall functions. Again the only changes occur in the first five rows of the Ajs matrix, Eq. (2.3.20a). Of the five boundary

conditions at the wall, the first three are written in the order given by Eqs. (2.3.3a,b,c) and the fourth and fifth are given by ko = 0

(2.4.15a)

CO = 0

(2.4.15b)

Sko = (r4)0 = 0

(2.4.16a)

6co = (r5)o = 0

(2.4.16b)

or in linearized form

The structure of the other matrices remains the same but, of course, the coefficients of the linearized momentum, kinetic energy and dissipation equations, Eqs. (2.3.24) to (2.3.26), respectively are different than those for k-e model equations without wall functions. These coefficients naturally vary depending on the wall functions used. The A0 matrix for the k-E model equations with wall functions, with the last three rows identical to those in Eq. (2.4.14), is

A0 =

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

0

-2v/Rx wo

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0 -1-z 0 0 0 0 -1- . 0 0 0 0

0

0

0

0

-1 _h

0

0

(2.4.17)

2. Solution of the CS and k-e Model Equations

34

References [1]

Cebeci, T. and Cousteix, J.: Modeling and Computation of Boundary-Layer Flows, Horizons Publishing, Long Beach, CA and Springer-Verlag, Heidelberg, Germany, 1998.

Computer Program for CS and k Models

3.0 Introduction In this chapter we present and discuss a code for two-dimensional incompressible

turbulent flows. The code, given in the accompanying CD-ROM, includes a computer program employing the CS model discussed in Section 1.1 and another program employing the k-e model discussed in Section 1.2. It is prepared in five

parts. In Part 1, discussed in Section 3.1, the components of the computer program common to both models are described. Part 2, discussed in Section 3.2, describes the components of the CS model, and Parts 3 and 4 describe the k-e model and its solution algorithm in Sections 3.3 and 3.4, respectively. The computer program for the k-c model includes the zonal method with a combination of the CS model for the inner region and the k-e model for the outer region as discussed in Section 2.3. It also includes the solution of the k-e model equations with and without wall functions. Finally Part 5 includes a brief description of the basic tools such as integration, smoothing and differentiation, used in the computer program.

3.1 Part 1 - Components of the Computer Program Common to Both Models Part 1 includes a MAIN routine which contains the logic of the computations and five subroutines: INPUT, IVPT, GROWTH, GRID and OUTPUT. The following subsections present a brief description of each routine.

3.1.1 MAIN Here we first read in input data (subroutine INPUT) and generate the initial turbulent velocity profile (subroutine IVPT) and the eddy viscosity distribution

3. Computer Program for CS and k-e Models

36

for the CS model (subroutine EDDY), k-profile (subroutine KEINITK), e-profile (subroutine KEINITK). Since linearized equations are being solved, we use an

iteration procedure in which the solutions of the equations are obtained for successive estimates of velocity, kinetic energy, dissipation profiles with a subsequent need to check the convergence of the solutions. A convergence criterion based on Svo is used and the iterations are stopped when Sva V0

< 0.02 tt(

During this iteration procedure, we introduce an under-relaxation procedure for the iterations as described in MAIN. This is useful, especially with transport equation turbulence models. When the solutions converge, we also check to see whether the boundarylayer thickness, 77e, used in the calculations for that x-station is large enough so that the asymptotic behavior of the solutions is reached. If this is not the case, we call subroutine GROWTH. After the convergence of the solutions, the OUTPUT subroutine is called and the profiles which represent the variables such as fj, uj, v?, k?, sj etc. are shifted.

3.1.2 Subroutine INPUT In this subroutine we read in input data and set up the flow calculations according to the following turbulence models listed below. CS model Huang-Lin k-e model Chien k-F model zonal method high Re # k-E model

In some problems, like airfoil flows, it is convenient to read in the dimension-

less airfoil coordinates x/c, y/c rather than the surface distance required in the boundary-layer calculations. In all calculations, the external velocity ue(x) either dimensional or dimensionless, ue/uoo, and freestream or reference velocity, uoo(uref), kinematic viscosity v (CNU), reference length c (chord), variable ?7-grid parameter K (VGP) discussed in subroutine GRID must be specified together with Ro (RTHA) and e f (CFA) needed to generate the initial turbulent velocity profile with subroutine IVPT. The input also requires the specification of the first grid point needed in the r7-grid generated by subroutine GRID. This is done by inputting yo (YPLUSW). defined by Y tLr V

3.1 Part 1 - Components of the Computer Program Common to Both Models

37

where v, (UTAU) is the friction velocity, ue c f/2 and yo is the variable grid parameter h1, discussed in subroutine GRID. Its typical values for CS, zonal and high Reynolds number k-e models are around 0.5 to 1.0. For low Reynolds number k -E model, values of yo around 0.10 to 0.50 are typical. In the present program, K is set equal to 1.12, yo equal to 0.5 for low Reynolds number k-e model and 1.0 for zonal and CS models. Since equations use transformed variables where y is given by

y=

vx/ue?7

and since the location of x where the turbulent flow calculations are started, x1, can be an arbitrary distance, in this subroutine we calculate xo in order to control yo better. The calculation of the pressure gradient parameter m(x) (P2) in the transformed momentum equation is achieved from the given external velocity ue(x) distribution and from the definition of m. The derivative of due/dx is obtained by using subroutine DIFF3 given in Section 3.5.

3.1.3 Subroutine IVPT This subroutine is used to generate the initial turbulent velocity profile for both models by specifying a Reynolds number based on momentum thickness, Rp = ue0/v and local skin-friction coefficient e f [= r,,,/jOue]. It makes use of Eq. (1.2.9) for y+ < 50; and Eq. (1.2.10) for y+ > 50. In terms of the Falkner-Skan variables, Eq. (1.2.9) can be written as elr7

77 <

el + c21n(el77) + c3[ln(e1q)]2 + ca[ln(e117)J3

- < < 50

el

4

e1

el

where cl, c2, c3 and C4 are the coefficients of Eq. (1.2.9) and

e1 = R y2c f '

V = RLi Rx = uex

RL =

uOOL

v

(3.1.2)

Similarly, for 712! 50/el, we can use the expression given by Coles with Granville's correction for the whole profile (1), now with 7 = y/S, u+ = k In y+ + c+ k [17(1 - cos 7r r7) + (772 - r73)J

(3.1.3a)

Here 17 is a profile parameter which is a constant equal to 0.55 for flows with zero pressure gradient provided that the momentum thickness Reynolds number Rp is greater than 5000. In terms of transformed variables, Eq. (3.1.3a) can be written as

3. Computer Program for CS and k-e Models

38

ef

!r

ln(e171) + c +

II

!,

)+

- cos 7r

3

rte

17e

77e

where

() - ` 1) 2

(1

_

7le=V"f 66

(3.1.3b)

(3.1.4)

It is clear that a complete velocity profile for a turbulent boundary-layer can be obtained from Eqs. (3.1.1) and (3.1.3) provided that the boundary-layer thickness 6 and the profile parameter 17 are known. Since they are not known at first, they must be calculated in a manner they are compatible with Re and c f.

A convenient procedure is to assume 6" (v = 0) and, calculate 17 from Eq. (3.1.3a) evaluated at the boundary-layer edge, 77 = r/e. The initial estimate of 6 is obtained from the power-law relation,

B_

n

(1+n)(2+n)

6

which for n = 7 0.10

Here 0 is calculated from the specified value of Re,

9= 71evRe

(3.1.5)

The next values of 6" (v = 1, 2, ... , n) are obtained from (3.1.6)

{d} where, with 01 ( Ra

=e

Ra/R6) given by [1]

\ \2 77 ( _) (1.9123016 + 3.056017 + 1.5172) (12 +

(3.1.7)

and 0 denoting the momentum thickness calculated from Eq. (3.1.5),

0 =6-601 dO

d6

and with

_01

_

(3.1.8a)

d 1- 6 dI7 d6

-

(3.1.8b)

d771 d6

(3.1.8c)

26

obtained by differentiating

with respect to J.

2

ue

Cf

uT

=

1

h:

/

\ \ V ue/I+

t in l hue

7aT

2171

+c 1

(3.1.9)

3.1 Part 1 - Components of the Computer Program Common to Both Models

39

3.1.4 Subroutine GROWTH For most laminar-boundary-layer flows, the transformed boundary-layer thickness qe(x) is almost constant. A value of 77e = 8 is sufficient. However, for turbulent boundary-layers, 77e(x) generally increases with increasing x. An estimate of 77e(x) is determined by the following procedure. We always require that 77e(xn) > r7e(xn-1), and in fact the calculations start

with 77e(xe) = 77e(x1). When the computations on x = xn (for any n > 1) have been completed, we test to see if [v fl < e at 77e(xn) where, say s = 5 x 10-4. This test is done in MAIN. If this test is satisfied, we set 77e(xn}1) = 77e(xn). Otherwise, we call GROWTH and set Jnew = Jold + t, where t is a number of

points, say t = 1. In this case we also specify values of (fjn, uj , vj, bj , k,,

e,' etc.) for the new 17j points. We take the values of uJ = 1, v = 0, f _ qjn=0. (77j-77e)uj+f,, k,' =k",JEj'J =e7b, J sn=0, J

3.1.5 Subroutine GRID The solution procedure for both models requires the generation of a grid normal to the surface, 77-grid, and along the surface, x-grid. The latter requirement is satisfied by specifying locations with intervals which can be uniform or nonuniform. Its distribution depends on the variation of ue with x so that the pressure

gradient parameter m(x) in the momentum equation can be calculated accurately. To ensure this requirement, it is necessary to take small Ox-steps where there are rapid variations in ue(x) and where flow approaches separation. For laminar flows, it is often sufficient to use a uniform grid in the 77-direction. A choice of transformed boundary-layer thickness 77e equal to 8 often ensures that the dimensionless slope of the velocity profile at the edge, f"(77e), is sufficiently small (< 10-3) and that approximately 41 j-points satisfies numerical accuracy requirements. For turbulent flows, however, a uniform grid is not satisfactory because the boundary-layer thickness 77. and dimensionless wall shear parameter f,,,, are much larger in turbulent flows than laminar flows. Since short steps in i must be taken to maintain computational accuracy when fu, is large, the steps near the wall in a turbulent boundary-layer must be shorter than the corresponding steps in a laminar boundary-layer under similar conditions. A convenient and useful a7-grid, discussed in [1[ and used in this subroutine is a geometric progression having the property that the ratio of lengths of any

two adjacent intervals is a constant; that is, hj = Khj_1. The distance to the j-th line is given by the formula 7j

Ki =h1jK-1' ?=1,2,...,J K>1 1

(3.1.10)

There are two parameters: h1, the length of the first. 077-step, and K, the ratio of two successive steps. The total number of points, J, can be calculated by the formula (see subroutine RIYGRID):

3. Computer Program for CS and k-s Models

40

r11h, x 10"

Fig. 3.1. Variation of K with hl for different ale-values

ln[1 + (K - 1)(77e/hj)] + 1 InK

(3.1.11)

In most problems, calculations are performed by selecting hl and K and calculating the transformed boundary-layer thickness 71e. An idea about the number of points taken across the boundary-layer with the variable n-grid that uses those parameters for different i7,-values can be obtained from Fig. 3.1. For example, for h1 = 0.01, K = 1.10 and 77e = 100, the ratio of 77,/hl is 104, and the number of points across the boundary-layer is approximately 70. Here in this program, we recommend K to be 1.12 (subroutine INPUT) which is sufficient for most turbulent flow calculations. Of course, if desired, this number can be changed with guidance from Fig. 3.1.

3.1.6 Subroutine OUTPUT This subroutine prints out the desired profiles of the momentum, kinetic energy and rate of dissipation equations, such as fj, uj, vj. kj, ej as a function of q. It also computes the boundary-layer parameters, c f, S', 0, Rb and R0.

3.2 Part 2 - CS Model This part of the computer program which uses the CS model has five subroutines

in addition to those described in Part 1. They include subroutines COEFTR, SOLV3, EDDY, GAMCAL, and CALFA and are briefly described in the following subsections.

3.2 Part 2 - CS Model

41

3.2.1 Subroutine COEFTR The solution of the momentum equation, Eq. (2.1.16), is much simpler than the solution of the k-e model equations. Since this equation is third order, we have three first-order equations, the first two given by the first two equations in Eq. (2.2.1) and the third by Eq. (2.2.2). After writing the difference equations for Eqs. (2.2.1a,b) and linearizing them, we obtain

bfj - 6fj-1 -

hj 2

(buj +buj_1) = (rl)j

buj - 6uj_I - 2- (6vj + bvj_1) = (r3)j_1

(3.2.1a)

(3.2.1b)

where

(r1)j = fj-i - f

(3.2.2a)

Itju2 1/2

(r3)j-1 = u()_1 - u(`) + hjvjvll1i2

(3.2.2b)

The third equation is given by

(SI)jbvj + (82)jbVj_1 + ($3)j6fj + (34)jbfj-1 + (s5)jbuj + (36)j6uj-1 = (r2)j (3.2.2c)

with (r2)j given by Eq. (2.3.14). The linearized boundary conditions correspond to Eqs. (2.3.4a,b) at 77 = 0 and to buj = 0 at Ii = rlj. This system of equations, as described in subsection 2.3.1, is again written in matrix-vector form given by Eq. (2.3.15) with Aj, Bj and Cj matrices given by AO =

1

0

0

0

1

0

Aj

1

-hj /2

0

(s3)j

(35) j

(31)j

0

-1

-hj+1/2

0 -1 -hi/2

1<j<J-1 (3.2.3a)

1

Aj

-hi/2

(s 3)J (so)j

0

1

-1 -hh/2

0 ( Si)J

Bj =

0

0 (S4) j (s6)j (32)j 0

0

1<j<J

0

(3.2.3b) 0 0 Ci =-

0

0< j<J-1

00 0 0 1 -hj+I/ 2

(3.2.3c)

and 6j and r"j by bh

bj = buj bvj

(ri)j

l

r'j =

(r2)j

(r2)j

0 :5j < J

(3.2.4)

3. Computer Program for CS and k-e Models

42

The solution of the Eq. (2.3.15) is again obtained with the block elimination method described in subsection 2.3.4. This subroutine contains the coefficients of the linearized momentum equation given by Eqs. (3.2.2c), (3.2.1) and (2.3.14). Since the calculations are for turbulent flow only, these coefficients for the first two computed x-stations are slightly different due to the use of two-point backward difference formulas (see subsection 2.3.1) for the streamwise derivatives in the momentum equation. This is needed to avoid oscillations caused by the specified initial velocity profiles. At the third x-station, the calculations revert back to the central differences for the streamwise derivatives described in [1]. In this case the coefficients (s1)j to (s6) j and (r2) j are given by (51)j = h3

10' )

i + 2 fjvl + 2 fjLi/2

(52)j = -tai lbj-1 + 2 fj_i + 2 f3(-)1/2

(3.2.5a)

(3.2.5 b)

n

(53)5 = 2 vjvl +

(5 3 ) 3

-1

(r2)j = Rj-1/2

=

2

vjUl 1

+

2 vjv,1/2 C,

2

(3.2.5 c)

(3 2 5 d) .

.

Vjv11/2

(85)3 = -a2ujUl

(3.2.5 e)

(56)j =

(3.2.5 f)

-a2uj_1

h 1(bjuy(-) -bj2llvj l1)+ct1(fv)3_1/2 - a2(u2)jv)

1/2

n- I 2vjv,1/2) - an(vn-1 j-1/2 7f _1/2 - fj ll (3.2.6)

where

n-i + an [(fv)n-i n-1 =-Lj_1/2 Rj_1/2 j-1/2 - (Uu2)n-1 j-1/2 ] - mn Lj-11/2 =

f hs 1(bjvj

- bj-lvj-1) + mi(fv)j-1/2 + m(1 -

(3.2.7a)

(u2)j-1/2]}n-1

(3.2.7b)

a1=m1+a, a2=m+a

3.2 Part 2 - CS Model

43

3.2.2 Subroutine SOLV3 This subroutine is used to solve the linear system given by Eq. (2.3.15) with the block elimination method discussed in subsection 2.3.4. To describe the procedure for obtaining the recursion formulas used in this subroutine we first consider Eq. (2.3.30). Noting that the rj matrix has the same structure as Bj and denoting the elements of yj, by yik (i, k = 1, 2, 3), we can write rj as ('Yll)j ('Y12)j ('Y13)j (721)j (-Y22)j ('123)j

rj=

0

0

(3.2.8a)

0

Similarly, if the elements of Aj are denoted by aik. we can write 4j as [note

that the third row of 4j follows from the third row of Aj according to Eq. (2.3.30c)]

(all)j (a12)j (a13)j

4j

(a21)j (a22)3 0

-1

(a23)3

-hj+1/2

0<j < J-1

(3.2.8b)

and for j = J, the first two rows are the same as the first two rows in Eq. (2.3.30b), but the elements of the third row, which correspond to the boundary conditions at j = J, are (0, 1, 0). For j = 0, do = AO; therefore the values of (aik)o are (a11)0 = 1

(a12)o = 0

(a13)o = 0

(a21)o = 0

(a22)o = 1

(a23)o = 0

(3.2.9a)

and the values of ('Yik)l are 1

(1711)1 = -1

('Y12)1 = - 2h1

(713)1 = 0 (3.2.9b) 11

(721)1 = (84)1

('723)1 = -2 [(S)ul hl

('722)1 = (S6)1 + (-Y23)1

The elements of the Aj matrices are calculated from Eq. (2.3.30a). Using the definitions of Aj, rj and Cj_1, we find from Eq. (2.3.30c) that for j = 1, 2, ... , J,

(all)j = 1

(a12)j = - 2' - (-Y13)j

(a13)j = 21 (-Y13)j

(0!21)j = (83)j

(a22)j = (S5)j - ('Y23)j

(a23)j = (S1)j +

2j

('Y23)j

(3.2.10a)

To find the elements of the rj matrices, we use Eq. (2.3.30b). With 4j defined by Eq. (3.2.8b) and Bj by Eq. (3.2.3b), it follows that for 1 < j < J,

3. Computer Program for CS and k-E Models

44

(yll)j =

1 (a23)j-1 +

-

2'

('Y12)j =

h+ 2j

j [(2) (a21)j-1

- (a22)j-1]

(yli)j {(c12)j_12 - (a13)j-1J

J

/zlo J

/zo

(y13)j = I(y11)j(a13)j-1 + (-Y12)(a23)j-11/j2

(y21)j = {(S2)j(a21)j-1 - (S4)j(a23)j-1 + 2,

I(S4)j(a22)j-1 -

(S6)j(a21)j-ll} /do

(y22)j = 1 (S6)j ZJ - (s2)j + (y21)j 1(a13)j-1 - (a12)j-1 2? J } /dl (3.2.10b) ('Y23)j = (y21)j(0112)j-1 +

(y22)j(a22)j-1 - (86)j

do = (a13)j-1(a21)j-1 - (a23)j-1(a11)j-1 - 2? [(a12)j-1(a21)3-1 - (a22)j-1(a11)j-1l

41 = (a22)j-1 2- - (a23)j-1

To summarize the calculation of Pj and aj matrices, we first calculate aak from Eq. (3.2.9a) for j = 0, Nk from Eq. (3.2.9b) for j = 1, aik from Eq. (3.2.10a) for j = 1, then yik from Eq. (3.2.10b) for j = 2, a;k from Eq. (3.2.10a) for j = 2, then yjk from Eq. (3.2.10b) for j = 3, etc. In the second part of the forward sweep we compute wj from the relations given by Eq. (2.3.31). If we denote the components of the vector wj by (wl)j '15j = (w2)3 (w3)j

0:5j < J

(3.2.11)

Then it follows from Eq. (2.3.31a) that for j = 0, (wl)O = (rl)o

(w2)0 = (r2)0

(3.2.12a)

(w3)0 = (r3)o

and from Eq. (2.3.31b) for 1 < j < J,

(wl)j = (rl)j - (711)j(wl)j-1 - (y12)j(w2)j-1 -

(y13)j(w3)j-1

(w2)j = (r2)j - (' 21)j(w1)j-1 - (y22)j(w2)j-1 - (y23)j(w3)j-1 (w3)j = (r3)j

(3.2.12b)

In the backward sweep, Ej is computed from the formulas given by Eq. (2.3.32).

With the definitions of Sj, dj and i9j, it follows from Eq. (2.3.32a) that

8uj = (w3)J

(3.2.13a)

3.2 Part 2 - CS Model

45

e2(all)J - el(a21)J

b71J = (a23)J(a11)J

(3 . 2 . 13b)

- (a13)J(a21)J

- (al3)JbyJ (3.2.13c)

bfJ = el (all)J

where el = (wl)J - (a12)JbuJ e2 = (w2)J - (a22)J6uJ

The components of b , for j = J - 1, J - 2, ... , 0, follow from Eq. (2.3.32b)

bv = (al l)j [(w2)j + e3(a22)j] - (a21)j (V11 )j - e3(a21)j (a12) j

(3.2.14a)

A2

buj=-hj bvj-e3

(3.2.14b)

b fj - (w1)j - (a12)jbuj - (a13)j6vj

(3.2.14c)

(a11)j

where ttj±1 bvj+i

e3 = (w3)j - buj+1 + A2 = (a21)j(a12)j -

h

2

h 2

1

- (a21)3(a13)j

(3.2.14d)

11(0122)1(all)j + (a23)j(all)j

To summarize, one iteration of Newton's method is carried out as follows. The vectors Tj defined in Eq. (2.3.15) are computed from Eqs. (2.3.14) and (3.2.2) by using the latest iterate. The matrix elements of Aj, Bj and Cj defined in Eq. (3.2.3) are next determined by Eq. (3.2.5a) to (3.2.5f). Using the relations in Eqs. (2.3.30) and (2.3.31), the matrices I'j and dj and vectors wj are calculated. The matrix elements for Tj defined in Eq. (2.3.30b) are determined from Eqs. (3.2.9b) and (3.2.10b). The components of the vector wi defined in Eq. (3.2.11) are determined from Eq. (3.2.12). In the backward sweep, the components of Sj are computed from Eqs. (3.2.13) and (3.2.14).

3.2.3 Subroutines EDDY, GAMCAL, CALFA These subroutines use the CS algebraic eddy viscosity formulation discussed in Section 1.1. In terms of transformed variables, (cm +)i and (em+)o are given by (£,;,)% = 0.16772

Rxv {1 - exp(-RxI4vjw 2/26/cn.)} ytr

(E.m)o = a R.(77J - fj) ttry where

(3.2.15a)

(3.2.15b)

3. Computer Program for CS and k-e Models

46

_

m 1/4 3/4 Rx v,.,,

(3.2.16)

Subroutine EDDY contains the expressions for the inner and outer regions. The intermittency expression used in the outer eddy viscosity formula, Eq. (1.1.8) is calculated in subroutine GAMCAL and the variable a given by Eq. (1.1.9) in subroutine CALFA.

3.3 Part 3 - k-e Model The structure of the k-e model, which includes the zonal method and the model for low and high Reynolds number flows, is similar to the CS model described in the previous section. It consists of the subroutines described below.

3.3.1 Subroutines KECOEF, KEPARM, KEDEF and KEDAMP Again we need a subroutine for the coefficients of the linearized equations for momentum, turbulent kinetic energy and rate of dissipation. We also need to generate initial profiles for the kinetic energy and rate of dissipation equations for both low and high Reynolds number flows. We do not need to generate the initial turbulent velocity profile for the momentum equation since it is already generated by subroutine IVPT discussed in Part 1. Then we need an algorithm, like SOLV3, to solve the linear system of equations for the zonal method and k-e model with and without wall functions for low and high Reynolds number flows.

To simplify the coding and discussion and the application of this computer program to other turbulence models, we use three additional subroutines to define the coefficients of the linearized equations for momentum, kinetic energy

and rate of dissipation given in subroutine KECOEF. The first of these three subroutines is subroutine KEPARM, which calculates the parameters b1, b2, b3 and production and dissipation terms and their linearized terms such as ( )7",

(

)

n

71

(

n

, (a )j , (a _&T ),

n

, ( -87v ) ;etc. in the equations for kinetic energy and rate

of dissipation.

The second of these three subroutines is subroutine KEDEF, which calculates D, E, F terms, (see Section 1.2), and their linearized terms such as (6)f , (W) , etc. in the k-E model associated with low Reynolds number effects, which in the present program correspond to the models of Huang-Lin and Chien discussed in Section 1.2.

The third of these subroutines is subroutine KEDAMP, which calculates near-wall damping terms f1, f2, ff, al., ce and their linearized terms which are for low Reynolds numbers and are model dependent.

47

3.3 Part 3 - k-e Model

The linearized coefficients of the momentum equation in subroutine KECOEF use both two and three point backward finite-difference approximations for the streamwise derivatives. For j < js, the coefficients (s1)j to ($6)j are given by Eq. (3.2.5) for the CS model. At j = js (Em)CS = (em)k-e

and (s7)j to (512)j are given by the following equations,

(s7)j = hj_1vj 8k 01

-hj-'vj-l f

(3.3.1a)

j

)j_l

(3.3.1b)

(8b)AS-

1

(3.3.1c)

-h.7_1 vj_1

/8b\

(3.3.1d)

8s Jj_1 )j

1v-

h-

(3.3.1e)

8E j

-h1vj-1

ab (3.3.1f)

8E j_1

for the k-E model. This subroutine also presents the coefficients of the kinetic energy equation,

(al)j to (a12)j and (r4)j in Eq. (2.3.25) and the coefficients of the rate of dissipation equation, (31)j to (,614)j and (rr5)j in Eq. (2.3.26). To discuss the procedure for obtaining the coefficients of the kinetic energy and rate of dissipation equations, consider Eq. (2.2.3). With x-wise derivatives represented either by two- or three-point backward differences, the finite difference approximations to Eq. (2.2.3) are

hj 1[(b2s)j - (b2s) 11 + mi (,f

-Qj-1/2 + F?

2mn(uk)j-1/2 + !j (Ok)n

f

1/2 = In `u9-1/2 `O2

- S _1/2

j-1/2

1/2

(8fln

j1/2J

Linearizing, we get

n

h?1

[(b2)8j+n ()okj +s j

(X21 6cj-(b2)?_l6sj_1

(Oba)n

- s _,

1n

6kj_, - sj-1

j

(-_ 8b 21

1

1

(3.3.2)

3. Computer Program for CS and k-E Models

48

vt

+2

(ak)

\n / n bkj+(aP)bej+I OVJ bv n

(81n

} (8p)a-1 bkj_i + ME ) j-i 6sj-1 + +

-1

bvj-1

+ 2 (fj,6sj+sjbfj+fJ_16sj-1+sj-l6fj-1) -m'[uubkj +kjbuj +ujI-li6kj_1+k? l6uj_1]

- 1 f (-)j bkj + (a,- j kj + (N)j_16kj-l -I

(19E

+(8e)j66j +(8k)j-lbkj-i+\8s

+2 [(8k)j6kj xn

+

l

)j-16E2-i]

/j

8k

{ (buj+buj-1) (8x)j-1/2+

[8k (8x )j bkj

8 (ak)n

(Of)"

+ 8k

8x

1

Skj-1 - (bsj + Ssj_1) 1

f

- 8j_1/2 r[af

(8x)n6fj

+ 8f

ax s-1/2

(2f)'j-1

8x6fj-1

I

(r4) j

(3.3.3)

The coefficients of Eq. (2.3.25) can be written as n

n2

(a1)j = 21 sj +

j

(az ) =

2j mi

(as) j

s

(

S

2 j-1/28 n f

(af )n 8x

S

= -mnk -

2n

(3.3.4a)

ax j

1/2 8ff

xn

-1 +

7a

(3 3 4b)

)n (3 3.4c)

(

.

x j-1/2

j, - xn a k)

(a4 )j = -mnk

1

j

1

(

2

(3 3 4d) .

n (3 3 4e) .

1

+2

(OF)n

(3.3.4f)

n a

IP) muj + 2 (c)3 xn n

.

n

(as)3 = 2

(017)j = hj isi (8k )

.

ax j-1/2

2

(a s ) =

.

.

-1

8

akj - 2 j-1/28k

8k n

().

1

8Q

2

8k (3.3.4g)

3.3 Part 3 - k-e Model

49

z (8P)

1

(as).i = -h;sj_1 (8b2)n ak i-1 + 2 8k 1 (8Q )n 2

(8k )j-1 + 2 7kj-1 m" mn

(alo)3 _ -h, 1(b2)j-1 + 21 fi-1 +

In

xn

T

(0b2)fl 9-1

8 n (af )9-1/2

xn

(88x

2

(3.3.4i)

)n

(3.3.4])

9-1/2

Qn

[(,) n

(3.3.4k)

-(8E) 7+(8E)

(8s) 7 +2 (12)J

(3.3.4h)

8 (0k )n

2 uj-1/28k 8x).i_1

(a9)i = hj 1(b2)3 + 21 f.Y +

n

- mnujn

xn

OF I n

1

_1

f[ 8E) j-1 + 2 (8P `

8Q

8F n l

8E) j-1 +

8E) j-1 (3.3.41)

(r4).7 = -[(b2s)7 -

[P 1/2 - Q 1/2 + Fj

+ ,in

1/2 - 2mn(uk)3-1/2

8f n

8k n

+xn 1Uj_ 1/2 (FX).?-1/2 - Si-112

(3.3.5a)

()j_1/2]

Remembering the definitions of the diffusion, production, dissipation, convection and F terms, Eq. (2.9.4a) can also be written as

(r4)j = -[diffusion + production - dissipation + F - convection]

(3.3.5b)

The parameter P, Q and F are model-dependent. As a result, the derivatives with respect to k, E and v will be different for each model. The derivations of Ok and i with respect to k and f are straightforward. In term of transformed variables the parameters P, Q and F in Huang and Lin's model, for example, are (here a is e) z

P=c,f,,RxkEv2

(3.3.6a)

2

Q=e+216k,

2

D

2

(3.3.6b)

r

F = 12Q (ss'

-2T

ll'

(3.3.6c)

We now consider the rate of dissipation equation given by Eq. (2.2.4). Following a procedure similar to the one used for the kinetic energy equation, the finite-difference approximation for Eq. (2.2.4) is

3. Computer Program for CS and k-e Models

50

-

(E)

hj l[(b3q)j - (b3q)j 1] + (Pl)7_1/2 + m1(fq)7 1/2 - (3mn - 1)(ue)7-1/2

_xu

1/2

(3.3.7)

a j 1/2(axj-1/2q1/2 (O"f

[--

x)j-1/2

After linearization, the resulting expression can be expressed in the form given by Eq. (2.3.26),

ml n

(L)n

n

2 qj + xn

f

2qj-1/2

(3.3.8a)

ax

(L)n mi n xn n {p2)j = 2 qj-1 + 2 q-1/2aff axj-1 n

n

(/33)j = --2 (ax //I j-1/2 -

-- (-/ xn 2

ax j_1/2

(3.3.8c)

(3m - 1)ej

(3.3.8d)

1

2

)n

1

2av

(as)j=(8Pl

(3.3.8e)

(pp,)n

1

- 1)e

2

-

asl n

('66)j _

1(3m

(3.3.8b)

(3.3.8f)

av

2

j

j- 1/2

(OQ,)n (a7)j = h.ilqjn

(68)j = -hilqj

(ak)j+(aklj 1 (,y,)n

ab3 n 1

a j

2

(ak )j-1 + 2

(011)2=h?lqjn(*)

ak

2("'

In

(3.3.8g)

1 (2Q,)n

j_

ak

2

j-1

DE' )j+()

(3.3.8h)

I

(3.3.8i)

- (3mn - 1)un - 2 u7_1/2C8 (ax/ {Q12)j = -h? lq7 1

(*)n

)j-1

I

+

2

- (3mn - 1)u 1 - 2 u7-1128- R 1

hj (b3)j + R

((314)j = -h11(b3)7_j +

n

ml

m1n

2

n

09

1

(3.3.8j)

/j n

2

fn +

fj

1+

(3.3.8k)

-X

2

xn 2

j_1/2 (aaxf

n (LaXf)

j-1/2

(3.3.81)

3.3 Part 3 - k-e Model

51

(r5)7 = -h,-1[(b3q)j - (b3q)j 1[

- P07-1/2 - (Q')j 1/2 + Ej 1/2 +,M11 (fq)j_1/2 - (3mn - 1)(tE)' 1/21 (r3E

[ [also

lug 1/2

`

(3.3.9a)

(8 f n

n

X)9-1/2

- qj 1/2

be `written are

Equation (3.3.9a) can

(r5)j = -[diffusion + generation - destruction - convection + EJ

(3.3.9b)

3.3.2 Subroutine KEINITK This subroutine generates the initial k-profile for low and high Reynolds numbers as well as the profile for the zonal method. For high Reynolds number flows, the kinetic energy profile k is determined by first calculating the shear stress r from

T = (Em ) cs

c0u

(3 3 10) .

8y

.

and using the relation between T and k,

k=T

(3.3.11)

a1

with al = 0.30. The calculation of r is easily accomplished in subroutine IVPT once the initial velocity profile is generated in that subroutine. For low Reynolds number flows, we assume that the ratio of r+/k+ is given by

k+

=

a(y+)2 + b(y+)3

y+ < 4.0

(3.3.12a)

C1 + C2Z + C3Z2 + C4Z3

60 ..5 y+ < 4.0

(3.3.12b)

y+ > 60

(3.3.12c)

1 0.30

where z =1n y+. The constants in Eq. (3.3.12a) are determined by requiring that at y+ = 4, k+ = 0.054,

/

(k+)/ = 0.0145

(3.3.13)

according to the data of [2). The constants c1 to c4 in Eq. (3.3.12b) are taken as Cl = 0.080015, c3 = 0.07821,

c2 = -0.11169 c4 = -0.0095665

(3.3.14)

3. Computer Program for CS and k-e Models

52

3.3.3 Subroutine KEINITG In this subroutine the rate of dissipation profile e is determined by assuming z

(Em)CS = (Em)k-e

fAcµ

(3.3.15)

or from e

f1 cµk2 (Em)CS

where (em)CS is determined from the CS-eddy viscosity model in subroutine EDDY. Whereas f,, is constant for high Reynolds number flows with a typical value of 1.0, it is not a constant for low Reynolds number flows. Its variation differs according to different models developed close to the wall, say y+ < 60 [3].

3.3.4 Subroutine KEWALL This subroutine provides the wall boundary conditions for the k-e model which

includes low (with wall functions) and high Reynolds number (without wall functions) flows as well as the zonal method. For low Reynolds numbers, there are four physical wall boundary conditions and one "numerical" boundary condition. They are given by Eqs. (2.1.13) and (2.3.3c). The last one is a numerical boundary condition and follows from the definition of uT (see Eq. (2.3.1a). For high Reynolds numbers, the "wall" boundary conditions are specified at a distance yo = (v/ur)yO . In this case we have a total of five boundary conditions. The physical boundary conditions are given by Eqs. (2.2.5), (2.2.6), and (2.2.7). The numerical boundary condition is given by Eq. (2.4.8).

3.3.5 Subroutine KESOLV This subroutine performs both forward and backward sweeps for low and high Reynolds numbers, including the zonal method, by using the block elimination method. When the perturbation quantities um(1, j) to um(8, j) are calculated so that new values of fj, uj, vj, etc., are calculated, a relaxation parameter rex is used in order to stabilize the solutions. In this subroutine, for the zonal method we also reset k, e in the inner region only. Since the CS model is used for the inner region, there is no need for these quantities. For safety, they are arbitrarily defined in this region.

3.4 Part 4 - Solution Algorithm When the system of first-order equations to be solved with the block elimination method becomes higher than, say 6, the preparation of the solution algorithm

3.4 Part 4 - Solution Algorithm

53

with the recursion formulas described in subroutine SOLV3 becomes tedious. A matrix-solver algorithm (MSA) discussed here can be used to perform the matrix operations required in the block elimination method. This algorithm consists of three subroutines, namely, subroutines GAUSS, GAMSV and USOLV. To illustrate its use, we discuss the replacement of SOLV3 with MSA. (1) Read in DIMENSION DUMM(3),BB(2,3),YY(3,81),NROW(3,81),GAMJ(2,3,81). AA(3,3,81),CC(2,3,81) DATA IROW.ICOL,ISROW,INP/3,3,3,81/

Here IROW, ICOL correspond to number of maximum rows and columns respectively. ISROW denotes the number of "wall" boundary conditions and INP the total number of j-points in the 77-direction, and

BB=Bj, YY=tVj, GAMJ=Fj, AA=A3, CC=Cj The first and second numbers in the arguments of AA, BB, CC and GAMJ correspond to the number of nonzero rows and columns in Aj (or Aj), Bj, Cj and Tj matrices, respectively (see subsection 2.3.4, for example). Note that Bj and Tj have the same structure and the last row of Bj and the-first two rows of Cj are all zero. The index 81 in YY, NROW, GAMJ, AA and CC refers to INP.

(2) Set the elements of all matrices Aj, Bj, Cj (and dj) equal to zero. (3) Define the matrices AD and CO by reading in their elements. Note that only

those nonzero elements in the matrices are read in since in (2) we set all the elements equal to zero. (4) Call subroutine GAUSS.

(5) Read in the elements of Bj and call subroutine GAMSV to compute I'1. (6) Define Aj according to Eq. (2.3.23b), call GAUSS and read in the elements of Cj.

(7) Recall the elements of Bj and call GAMSV to compute r2.

(8) Repeat (6) and (7) for j < J. (9) At j = J, read in the last row of A,j which is also equal to the last row of J. (10) Compute w'o according to Eq. (2.3.31a). Here TD = RRR(1,81).

(11) Define the right-hand side of Eq. (2.3.31b) and compute tVj according to Eq. (2.3.31b).

3. Computer Program for CS and k-e Models

54

(12) In the backward sweep, with Sj corresponding to UM(I,J), compute b'j according to Eq. (2.3.32a) by calling USOLV at INP.

(13) Define the right-hand side of Eq. (2.3.32b) and solve for 6j by calling

USOLV for j = J- 1,J-2,...,0. This algorithm is very useful to solve the linear system for the k-e model equations. With all Aj, Bj, Cj matrices and r j nicely defined in subroutine KECOEF, the solution of Eq. (2.3.15) is relatively easy.

3.5 Part 5 - Basic Tools This part of the computer program includes basic tools to perform smoothing, differentiation, integration and interpolation. For example, subroutine DIFF3 provides first, second and third derivatives of the input function at inputs. First derivatives use weighted angles, second and third derivatives use cubic fits. Subroutine INTRP3 provides cubic interpolation. Given the values of a function (Fl) and its derivatives at N1 values of the independent variable (XI), this subroutine determines the values of the function (F2) at N2 values of the independent variable (X2). Here X2 can be in arbitrary order. Another subroutine used for interpolation is subroutine LNTP; it performs linear interpolation.

References

[2]

Cebeci, T. and Cousteix, J.: Modeling and Computation of Boundary-Layer Flows, Horizons Publishing, Long Beach, CA and Springer, Heidelberg, Germany, 1998. Mansour, N. N., Kim, J., and Moin, P.: Near-Wall k-e Turbulence Modeling. J. AIAA,

(3)

27,1068-10731 1989. Cebeci, T.: Analysis of Turbulent Flows. Elsevier, 2003.

(1)

Computer Program for CS Model for Flows with Separation

4.0 Introduction As discussed, for example, in [1], for a given external velocity distribution, the solutions of the boundary-layer equations are singular at separation. To continue calculations beyond separation, it is necessary to compute the external velocity or pressure as part of the solution. This procedure is known as the inverse mode. Catherall and Mangler [2] were the first to show that modification of the external velocity distribution near the region of flow separation leads to solutions free of numerical difficulties. By prescribing the displacement thickness or wall shear

stress as a boundary condition in addition to the usual boundary conditions, the boundary-layer equations can be integrated through the separation location and into the region of reverse flow without any evidence of singularity at the separation point [1].

A problem associated with using an inverse mode to solve the boundarylayer equations is the lack of a priori knowledge of the required displacement thickness or the wall shear. The appropriate value must be obtained as part of the overall problem from the interaction between the boundary layer and the inviscid flow. In the case of internal flow, the problem is somewhat easier because the conservation of mass in integral form can be used to relate pressure p(x) to velocity u(x, y) in terms of mass balance in the duct. For two-dimensional external flows, two procedures have been developed to couple the solutions of the inviscid and viscous equations. In the first procedure, developed by LeBalleur [3] and Carter and WtTornoin [4], the solution of the boundary-layer equations is obtained in the standard mode, and a displacementthickness, 6*°, distribution is determined. If this initial calculation encounters separation, then 6*°(x) is extrapolated to the trailing edge, and one complete cycle of the viscous and inviscid calculation is performed with the boundarylayer equations now solved in the inverse mode. This will, in general, lead to

56

4. Computer Program for CS Model for Flows with Separation

two different external velocity distributions, ue t(x) derived from the inverse boundary-layer solution, and uC2(x) derived from the updated approximation to the inviscid velocity past the airfoil with the added displacement thickness. A relaxation formula is introduced to define an updated displacement-thickness distribution, Lueti(x)

S* = S*0(x) { 1

l

I uea(x)

- 11 }

(4.0.1)

where w is a relaxation parameter, and the procedure is repeated with this updated mass flux. In the second approach, developed by Veldman [5), the external velocity ue(x) and the displacement thickness b*(x) are treated as unknown quantities, and the equations are solved in an inverse mode simultaneously in successive sweeps over the airfoil surface. For each sweep, the external boundary condition for the boundary-layer equation in dimensionless form, with ue(x) normalized

with u,,, is written as ue(x) = tL(x)+ Sue(x)

(4.0.2a)

where u°(x) denotes the inviscid velocity and Sue the perturbation due to the

displacement thickness, which is calculated from the Hilbert integral

if

Xa

Sue(x) =

_

(ues*) xdaor

d T-7

7r

(4.0.2b)

Sa

The above expression is based on the thin airfoil approximation with the term d/da(ueS*) denoting the blowing velocity used to simulate the boundary layer in the interaction region (xa, xb). This approach is more general and will be used in the present approach for calculating boundary-layer flows with separation. We consider a laminar and turbulent flow. We assume the calculations start at the leading edge, x = 0, for laminar flow and are performed for turbulent flow at any x-location by specifying the transition location. The use of the two-point finite-difference approximations for streamwise derivatives is proper and does not cause numerical difficulties if there is no flow separation. If there is one, then it is necessary to use backward difference formulas as discussed in subsection 2.3.1.

We employ two separate but closely related transformations. The first one is the Falkner-Skan transformation in which the dimensionless similarity variable al and a dimensionless stream function f (x, 77) are defined by Eqs. (2.1.1). It is a convenient and useful transformation to generate initial conditions which start either as a flat plate flow or stagnation flow. It is also useful to perform the boundary-layer calculations subject to the boundary conditions that correspond to the standard mode (external velocity distribution given). The resulting equations from this transformation are given by Eqs. (2.1.3).

57

4.0 Introduction

In the inverse mode, since ue(x) is also an unknown, slight changes are made to the transformation given by Eq. (2.1.1), replacing ue(x) by u"O and redefining

new variables Y and F by

Y=

u./vx y, V)(x, y) = u vx F(., Y),

=L

(4.0.3)

The resulting equation and its wall boundary equations can be written as (bF'")'

FF" _

+

Y=O,

aF' ('b1 - F" l

F'=0,

w dw

F=O

(4.0.4)

(4.0.5)

Here primes denote differentiation with respect to Y and w = ue/uoo . The edge boundary condition is obtained from Eq. (4.0.2). By applying a discretization approximation to the Hilbert integral, Eq. (4.0.2b), we can write (see subsection 4.2.3) N

i-1

CiiDi + ECijDj + > CijDj

ue(xi) =

(4.0.6)

j=i+1

j=1

where the subscript i denotes the c-station where the inverse calculations are to be performed, Cij is a matrix of interaction coefficients obtained by the procedure described in subroutine HIC, and D is given by D = ueS*. In terms of transformed variables, the parameter D becomes

D=

_

RL (Yew

- Fe)

(4.0.7)

and the relation between the external velocity ue and displacement thickness S* provided by the Hilbert integral can then be written in dimensionless form as Y = Ye,

)(YFe(e1) - Fe(el)] = gi

(4.0.8)

where (4.0.9a)

A = Cii e}/RL

_

i-1

N

gi = ue(t) + E CijDj + j=1

CijDj j=i+1

(4.0.9b)

4. Computer Program for CS Model for Flows with Separation

58

4.1 Numerical Method The numerical method for the inverse problem is similar to the numerical must method described for the standard problem in Section 2.3. Since be computed as part of the solution procedure, we treat it as an unknown. Remembering that the external velocity w is a function of 6 only, we write

w=0

(4.1.1)

As in the case of the standard problem, new variables Y), V(, Y) are introduced and Eq. (4.0.4) and its boundary conditions, Eq. (4.0.5) and (4.0.8), are expressed as a first-order system,

F'=U

(4.1.2a)

U'=V

(4.1.2b)

(bV)'+ IFV = (U aU - V OF) -Cwdww

(4.1.2c)

Y=0, U=w,

F=U=O

(4.1.3a)

(4.1.3b) \F+(1- 1Ye)w=9i Y=Yei Finite-difference approximations to Eqs. (4.1.1) and (4.1.2) are written in a

similar fashion to those expressed in the original Falkner-Skan variables, yielding

hj1(wjn - wj 1) = 0

(4.1.4a) 1/2

(4.1.4b)

h371 (Uj,, - UU 1) = V3 1/2

(4.1.4c)

h1 1(F' - F-1) = Uj

hj-1(bjnVj" - bjn'1)+{2+a"//)(FV)j_1/2

+ an [(w2)j-1/\\2 -FLARE 3-1/21 + a"(V." 1/2Fj 1/2 - F 1/2Vi 1/2) =

(4.1.4d)

where

Rj-1/2 = -LL-1/2 + an[(FV) i/2 - FLARE(U2)j-1/2] Lr-1

9-1/2

-

rh-1 l

j (Jb 7y- b _1v? _ 1) + 1(FV)j-1/2 7

(4.1.5a) a"(w2)j-1/2]n-1

(4.1.5b)

In Eq. (4.1.4d), the parameter FLARE refers to the Flugge-Lotz-Reyhner approximation [1] used to set uyy equal to zero in the momentum equation wherever u < 0. As a result, the numerical instabilities that plague attempts to integrate the boundary-layer equations against the local direction of flow are

4.1 Numerical Method

59

avoided. In regions of positive streamwise velocity (uj > 0), it is taken as unity and as zero in regions of negative streamwise velocity (u3 < 0). The linearized form of Eqs. (4.1.4) and (4.1.3) can be expressed in the form given by Eq. (2.4.15) or 6U0

6F0 b.c. b.c.

:

6V0

6.0

6Fj

6Uj

0. 0

6Fj

6wj

6Vj

6Uj

6Vj

6wj

.................................................. O 0 0 1 0 0

:

6F0

(rl)0

0

1

0

0

0

0

0

0

6UG

(r2)0

0

-1

-21

0

0

1

=41

0

6VG

(r3)0

0

0

0

-1

0

0

0

1

6w0

(r4)0

...................................}................ f-.h

0

1

0

0

0

0

0

0

0

6Fj

(rl)j

(31)j

(c7)1

0

0

0

0

6Uj

(r2)j

0

0

0

6 vj

(r3)j

6wj

(r4)j

0

6Fj

(rl)J

(33)J (35)j (cl)J

(37)J

6Uj

(r2)J

1

(s8)j; (53)j

(34)j

('c)j

('2)j

(s5)3

0

0

0

0

0

-1

0

0

0

0

0

0

0

-1

-1

-jj

0

0

(32)J

(ce)J

0

0

71

0

0

72

6Vj

(''3)J

0

0

0

1

0

-1

6wj

(r4)j

0 0 0 1 ......................... ..........................

(34)J (3G)J 0 0

b.c.

0

0

b.c.

0

1

(4.1.6)

with Sj and rj now defined by 6Fj

(ri)j

SUj bj = 6Vj Swj

rj =

,

(r2)j

(4.1.7)

(r3)j (r4)j

and Aj, Bj, Cj becoming 4 x 4 matrices defined by

Ao =

1

0

0

0

0

1

0

0

0-1-2 0 00 1

AJ =

0

-1

-2

0

-2 0 -1 -

1 ,

Aj =

(s3)j (s5)j

0

0

0

0

(S1)j

-1

0

2

(s3)J (s5)J (sl)J (s7)J

(4.1.8a)

0

0

(84)j (S6)j (s2)j (88)j

Bj =

,

1<j<J-1

,

0

-1 -3

0

(s7)j

71

0

0

72

0

0

0

0

0

1

0

-1

0

0

0

0

1<j<J (4.1.8b)

Cj =

00 00

0

0

0

0

0 1- 2 0

,

0<j<J-1

(4.1.8c)

00 0 1 Here the first two rows of A0 and CO and the last two rows of Bj and Aj correspond to the linearized boundary conditions,

4. Computer Program for CS Model for Flows with Separation

60

6F0 = 6Uo = 0;

5UJ - bwJ = wJ - Uj,

(4.1.9)

71bFJ + 72bwJ = 73

where

72=1-)YJ,

71=A,

73=gi-(71FJ+72wJ)

(4.1.10)

As a result

(rl)o = (r2)o = 0

(4.1.11a)

(4.1.11b) (r4)J = wJ - UJ The third and fourth rows of A0 and Co correspond to Eq. (3.2.1b) and the linearized form of Eq. (2.3.6a), that is,

(r3)J = 73,

bwj - bwj_1 = Wj-1 - wj = (r4)j-1

(4.1.12)

if the unknowns f, u, v are replaced by F. U and V. Similarly, the first and second rows of Aj and Bj correspond to Eq. (3.2.1a) and (2.3.6c) with two terms added to its left-hand side, (s7)j6wj + (s8)jbwj-1

(4.1.13a)

with (s7)j and (s8)j defined by

(S7)j = a'wj,

($8)j = anwj-1

(4.1.13b)

The coefficients (s1) j to (8r,) j defined by Eqs. (3.2.5) remain unchanged provided we set

a'1=2+a",

a2=an

(4.1.14)

and define (r2)j by

(r2)j = Rs-1/2 - [h?'(bjVj - bj-1Vj-1) + (2

+an)(FV)j-1/2

+ & [(w2)j_1/2 - FLARE(U2)j_1/2I

+ a"(V? 1/2Fj-1/2 - F i/2Vj-1/2))

(4.1.15)

The remaining elements of the rj vector follow from Eqs. (3.2.2), (4.1.12) and (4.1.15) so that, for l < j:5 J, (r1)j, (r2)3. (r3)j_1 are given by Eqs. (3.2.2a), (4.1.15) and (3.2.2b), respectively. For the same j-values, (r4)j_1 is given by the right-hand side of Eq. (4.1.12). The parameters 71, 72 and 73 in Eq. (4.1.9) determine whether the system given by the linearized form of Eqs. (4.1.4) and their boundary conditions is

to be solved in standard or inverse form. For an inverse problem, they are represented by the expressions given in Eq. (4.1.9) and for a standard problem

by-11 =0,72=1.0and 73=0. It should be noted that for flows with separation, it is necessary to use backward differences as discussed for the CS and k-e models in subsection 2.3.1.

61

4.1 Numerical Method

In that case, the coefficients (Si), to (se)j are given by Eq. (2.3.13), and (r2)j by Eq. (2.3.14) with the relations given by Eq. (4.1.15). The solution of Eq. (2.3.15), with bj and rj defined by Eq. (4.1.7) and with Aj, Bj and Cj matrices given by Eqs. (4.1.8), can again be obtained by the block-elimination of subsection 2.3.4. The resulting algorithm, similar to SOLV3, called SOLV4, is given in the accompanying CD-ROM.

Numerical Method for Wake Flows In interaction problems involving airfoils, it is usually sufficient to neglect the wake effect and perform calculations on the airfoil only, provided that there is no or little flow separation on the airfoil. With flow separation, the relative importance of including the wake effect in the calculations depends on the flow separation as shown in Fig. 4.1 taken from [1]. Figure 4.1a shows the computed separation locations on a NACA 0012 airfoil at a chord Reynolds number, Rc of 3 x 106. When the wake effect is included, separation is encountered for angles of attack greater than 100, and attempts to obtain results without consideration of the wake effect lead to erroneously large regions of recirculation that increases with angle of attack, as discussed in [1]. Figure 4.1b shows that the difference in displacement thickness at the trailing edge is negligible for a = 10° but more than 30% for a = 16°. As discussed in [1], the inverse boundary layer method described here can also be extended to include wake flows. This requires the specification of a turbulence model for wake flows and minor modifications to the numerical method. The extension of the CS model for wall boundary layers to wake flows is given by the following expressions described in [1]:

1.0

0.9 0.8

x/c 0.7 0.6 0.5

10

14

12

16

0.00 0.0

18

(a)

0.2

0.4

0.6

0.8

1.0

x/c

CL

(b)

Fig. 4.1. Wake effect on (a) flow separation and (b) displacement thickness - NACA 0012

airfoil. -, with wake; - - -, without wake.

62

4. Computer Program for CS Model for Flows with Separation

em = (Em)w + [(Em)t.e. - (Em)w] exp

-(x

(4.1.16)

ASt.e.

where 6t,e, is the boundary layer thickness at the trailing edge, A is an empirical parameter, (Em)t.e. is the eddy viscosity at the trailing edge, and (em)w is the eddy-viscosity in the far wake given by the larger of flmin

(em)y, = 0.064

f

and

(em)w = 0.064

(ue - u) dy

(4.1.17)

(ue - u) dy

(4.1.18)

00

Jvmn,

with ymin denoting the location where the velocity is a minimum. The studies conducted in [6] indicate that a choice of A = 20 is satisfactory for single airfoils. Calculations with different values of A essentially produced similar results, indicating that the modeling of wake flows with Eq. (4.1.16) was not too sensitive to the choice of A. The application of the above model to wake flows with strong adverse pressure gradient, however, indicated that this was not the case and the value of the parameter is an important one. On the basis of that study, a value of A = 50 was found to produce best results and is used in the present computer program. A modification to the numerical method of the previous section arises due to the boundary conditions along the wake dividing streamline. The new "wall"' boundary conditions on f and u now become

v0 =0

fo=0,

77=0,

(4.1.19)

so that the second row of Eq. (4.1.6) can de written as 0

0

1

0

(4.1.20)

Before the boundary-layer equations can be solved for wake flows, the initial velocity profiles must satisfy the wall and edge boundary conditions. When the calculations are first performed for wall boundary layer flows and are then to be extended to wake flows, it is necessary to modify the velocity profiles computed for wall boundary layers. This is done in subroutine WAKEPR of the computer program.

4.2 Computer Program In the accompanying CD-ROM, we present a computer program for obtaining boundary-layer solutions on airfoils and their wakes. The code can be used to solve both standard and inverse problems. For standard problems, it is similar to the computer program described in Section 3.2 except that it also provides

4.2 Computer Program

63

solutions to wake flows and is applicable to both laminar and turbulent flows. For inverse problems, it uses the numerical procedure described in the previous sections, and when coupled to an inviscid flow, it can also be used in inviscidviscous interaction problems, as described in Section 4.3. The computer program consists of a MAIN and 15 subroutines, INPUT,

IVPL, HIC, EDDY, SWTCH, COEF, WAKEPR, DIFF1, LNTP, INTEG, AMEAN, SOLV4, EDGCHK, CALFA and GAMCAL. MAIN, as before, is used to control the logic of the computations. Here the parameter 9i in Eq. (4.0.8) is also calculated, with i-1

SUM1 = F CijDj

(4.2.1a)

j=1 and

N

SUM2 = E CijDj

(4.2.1b)

j=i+1

The initial displacement thickness (6*) distribution needed in the calculation of Dj is computed in subroutine INPUT by assuming a 6* distribution based on a flat-plate flow and given by 6*

= 0.036 HR., 0.20

(4.2.2)

with H = 1.3. Of the 15 subroutines, subroutine WAKEPR is used to modify the profiles resulting from wall boundary layers for wake profiles. Except for this subroutine and except for subroutines INPUT, IVPL and HIC, the remaining subroutines are similar to those described in Sections 3.2 and 3.3. For this reason, only these three subroutines are described below.

4.2.1 Subroutine INPUT This subroutine is used to generate the grid, calculate 7tr in the eddy viscosity formulas, initial 6*-distribution, and pressure gradient parameters m and m1. The following data are read in and the number of j-points J(NP) is computed from Eq. (3.1.11). NXT NXTE NXS

NPT ISWPT

Total number of x-stations Total number of x-stations on the body NX-station after which inverse calculations begin Total number of i7-grid points Number of sweeps in the inverse problem. Usually a value of 10 is sufficient for low angles of attack; higher values are needed for high angles of attack

4. Computer Program for CS Model for Flows with Separation

64

RL

XTR ETAE VGP

DETA(1)

Reynolds number, u,,.c/v x/c value for transition location Transformed boundary layer thickness rle at x = 0, ETAE = 8.0 K is the variable-grid parameter. Take K = 1.0 for laminar flow and K = 1.14 for turbulent flow. For a flow consisting of both laminar and turbulent regions, take K = 1.14 Dii(ht) - initial step size of the variable grid system. Take hl = 0.01 for turbulent flows

P2(1)

m at x = 0 (NX = 1)

x/c, y/c

Dimensionless airfoil coordinates Dimensionless external velocity

ue/uoo

4.2.2 Subroutine IVPL At x = 0, Eq. (2.1.3) reduces to the Falkner-Skan equation, which can be solved

subject to the boundary conditions of Eq. (2.1.13a) and (2.1.14a). Since the equations are solved in linearized form, initial estimates of fj, uj and vj are needed in order to obtain the solutions of the nonlinear Falkner-Skan equation. Various expressions can be used for this purpose. Since Newton's method is used, however, it is useful to provide as good an estimate as is possible and an expression of the form. up

(177,

3 277e

l3

(4.2.3) 2

1

usually satisfies this requirement. The above equation is obtained by assuming a third-order polynomial and by determining constants a, b, c from the boundary

conditions from one of the properties of momentum equation which requires

that f"=0at77=77e. The other profiles fj, vj follow from Eq. (4.2.3) and are obtained by integrating and/or differentiating Eq. (4.2.3) for fj and vj, respectively.

4.2.3 Subroutine HIC This subroutine calculates the coefficients of the Hilbert integral denoted by C. While they can be generated from any suitable integration procedure, we use the following procedure which is appropriate with the box method (1). We evaluate

Hi

da

= Jg' G(a) V - a

where

G(a) =

dF da

(4.2.4)

4.2 Computer Program

65

Cn), except with F denoting any function, so that over each subinterval the two enclosing the point = i, we replace G(a) by its midpoint value:

=a d = Gn-1/2 Jan-1 V -a = Gn-1/21n £n

(s)

L

-1

do-

i - cn-1 Ci - n

(4.2.5)

Making the further approximation,

Gn-1/2 = we can write

Fn - Fn-1

n-

n-1

£n dF do = En(Fn - Fn-1) n-1 da - a

where forn54ior i+1 En'

i

=

(4.2.6)

n

the cancellation the Ci with the constant term, account should be taken of the linear variation of G from one interval to the next. Thus, we take the linear interpolation Gi-1/2(&1

G-

- Ci) + Gi+112W - &1) + 2(Gi+1/2 - Gi_112)(a - 0 ei+I - V-1

so that ci

&1 Gdo, _ G:-1/2(ea+1 - 0 + Gi+1/2W

In

(4.2.8)

- 2(Gi+1/2 - Gi-1/2)

Replacing the midpoint derivative values by difference quotients, we obtain 0+1 le-i

dF

-

doi;t -

= Eii(Fz . - Fa .- 1) + E?x+1 (Fx+l - Fa.)

where ei+1 _ i E8i

(4.2.9)

i - `-1

+2

In V

(4.2.10a)

Li-1 1nS

i-

ti-1 -2 (4.2.10b)

Thus

4. Computer Program for CS Model for Flows with Separation

66

Hi = Ez(F2-Fl)+E3(F3-F2)+...+ELL 1(FL-1 -FL_2)+EL(FL- F) = -E22F1 + (E2 - E3)F2 +... + (EL-1 - EL)FL-1 + ELFL (4.2.11)

so, finally the Cij of Eq. (4.0.6) are given by Ci? = 1(Ej' - E?+1) 7r

and the Ei are given by Eqs. (4.2.10a) and (4.2.10b) with Elf = EL+1

References [1] [2]

Cebeci, T., An Engineering Approach to the Calculation of Aerodynamic Flows. Horizons Publishing, Long Beach, CA and Springer, Heidelberg, Germany, 1999. Catherall, D. and Mangler K. W., The Integration of the Two-Dimensional Laminar

Boundary-Layer Equations Past the Point of Vanishing Skin Friction. J. Fluid Me[3]

[4)

[5] (6]

chanics, 26, 163, 1966. LeBalleur, J. C., "Couplage Visqueux-non Visqueux: Analyse du Probleme Incluant Decollements et Ondes de Choc," La Rech. Aerosp., 1977-6, 349. English translation in ESA TT476, 1977. Carter, J. and Wornom, S. F., "Solutions for Incompressible Separated Boundary Layers Including Viscous-Inviscid Interaction," in Aerodynamic Analysis Requiring Advanced Computers, NASA SP-347, p. 125, 1975. Veldman, A. E. P., New Quasi-Simultaneous Method to Calculate Interacting Boundary Layers. AIAA J., 19, 769, 1981. Cebeci, T. and Chang, K. C., "Compressibility and Wake Effects on the Calculations of Flow over High Lift Configuration," 48th Annual Conference, Canadian Aeronautics and Space Institute, Toronto, Canada, April 2001.

Hess-Smith Panel Method with Viscous Effects

5.0 Introduction Calculation of separating flows using boundary-layer theory requires inviscid and boundary-layer methods in which the calculations are performed in an iterative manner so that each successive inviscid flow solution provides the pressure distribution for the next boundary layer solution. The displacement (viscous) effect is then used to modify the inner boundary conditions for a near inviscid calculation; the procedure in this "cycle" known as the interactive boundarylayer scheme must then be repeated until convergence is obtained, see Fig. 5.1.

Boundary

Inviscid Method

Layer Method Viscous Effect

Fig. 5.1. Interactive boundary-layer scheme.

For incompressible flows, a panel method is an ideal inviscid method for this approach. Of the several panel methods, here we choose the one due to Hess and Smith [1]. After a brief description of the mathematical models for the interaction process in an inviscid flow (Section 5.1), we discuss the Hess-Smith (HS) panel method (Section 5.2). The procedure for incorporating the viscous effects into the panel method is discussed in Section 5.3. Changes required in an inviscid method to extend the viscous flow calculations into the wake of an

5. Hess-Smith Panel Method with Viscous Effects

68

airfoil are discussed in Section 5.4. A brief description of the computer program for the HS method with viscous effects is given in Section 5.5.

5.1 Mathematical Models for the Interaction Process in Inviscid Flows Viscous effects can be incorporated into an inviscid method by replacing the real flow with an equivalent fictitious inviscid flow such that the velocity components at the edge of the boundary layer are equal in both cases. We assume that in the fictitious flow, u = u,, for y < 6 . This fictitious flow can be modeled by using

either the "solid" displacement surface concept or the "transpiration" model concept proposed by Lighthill [2]. Figure 5.2a shows the real flow and Figs. 5.2b and 5.2c the equivalent fictitious flow using the displacement surface and blowing velocity concepts, respectively for two-dimensional flows.

u.

y=s

y=8

y

y=a

y=o

AN (a)

(b)

(c)

Fig. 5.2. Model of viscous effects in inviscid method in two dimensions (a) real flow, (b), (c) equivalent fictitious flow.

Both displacement surface and transpiration models are based on first-order boundary-layer theory, that is; the shear layer thickness is assumed to be thin and the transverse variation in pressure across the layer is assumed to be negligible. When this is not the case, it is necessary to use second-order boundary-layer theory, as discussed in [3]. To describe the mathematical models for each concept, it is convenient, following the discussion in (3], to work in terms of the difference between the two flows corresponding to the real viscous flow and an equivalent inviscid flow de-

fined as analytical continuation, or smooth extrapolation, of the inviscid flow outside the shear layers in the direction normal to the interface between them. For a two-dimensional compressible flow with Qu representing the real viscous flow and oiui the equivalent inviscid flow the difference of the continuity equations yields a (Qiui - Qu) + .-(Aivi - Qv) = 0

(5.1.1)

5.1 Mathematical Models for the Interaction Process in Inviscid Flow

69

Integrating across the boundary layer from y = 0 to 6 and representing the values at the wall by subscript w, we get 6

d

d

piwviw =

(Pi'ui - Pu)dy

(5.1.2)

0

since both (givi - ov) and (piui - pu) vanish at y = 6 because of the matching condition, while v = 0 at the wall (y = 0). If a displacement thickness, 6A is defined by 6

r

1

6A =

J

piwuiw

(piui - p'u)dy

(5.1.3)

0

then we obtain the blowing velocity defined by Lighthill, v:w

=

d (oiwuiw6A)

1

(5.1.4)

OW dx

The "classical" displacement thickness, 6B, is defined by the condition that the total mass flow in the equivalent inviscid flow shall equal that in the real viscous flow, so that Sp

Sr

piuidy 61

oudy

(5.1.5)

0

It is shown in [3) that y = 6B is indeed a streamline of the equivalent inviscid flow.

The relation between these two alternative definitions of displacement thickness can be obtained by adding

0

to both sides of Eq. (5.1.5). We obtain SB

f oiuidy = oiwuiw6A

(5.1.6)

0

Thus the transpiration condition given by Eq. (5.1.4) is equivalent to piwviw =

d d.L

f piuidy

(5.1.7)

0

which provides a physical explanation of it; that the mass flow injected normal

to the wall in the equivalent inviscid flow, in order to displace the dividing

5. Hess-Smith Panel Method with Viscous Effects

70

streamline outwards the required distance 6B from the surface, must be equal to the streamwie rate of change of the mass flow deficit in the boundary layer in the range 0 < y < 6B. In summary we see that there are two alternative inner boundary conditions which can be used in the calculation of the equivalent inviscid flow, namely, (1) a transpiration condition of nonzero normal velocity at the surface, Eq. (5.1.4), or (2) the condition that the displacement surface is a streamline, Eq. (5.1.5). Both mathematical models are equally valid and general representations of the displacement effect of the boundary layer on the external flow. In practice, however, the transpiration model is more convenient to use. For example, if a panel method is used to calculate the inviscid flow, the matrix of influence coefficients (Section 5.2) can be set up and inverted once and for all, so that subsequent inviscid calculations become trivial, while with a finite difference or finite volume method a fixed computational grid can be used throughout. For this reason, we will only use the transpiration model here.

Matching Conditions in the Wake With the assumptions inherent in first-order boundary-layer theory, the conditions that are needed to effect a matching between the viscous and inviscid calculations to simulate a turbulent wake with the transpiration model are similar to those discussed for the conditions on the airfoil surface. A dividing streamline is chosen in the wake to separate the upper and lower parts of the inviscid flow, and on this line discontinuities are required in the normal components of velocity, so that it can be thought of as a source sheet. At points C and D on the upper and lower sides of the dividing streamline (Fig. 5.3), the components of transpiration velocity, vj,,, and vii are, respectively, see Eq. (5.1.4), d oiu dx

Vi. = 1 (Qiuutxc6u) upper edge of viscous layer upper displacement surface

bu

' lower displacement surface `lower edge of viscous layer

Fig. 5.3. Notation for the airfoil trailing-edge region.

(5.1.8)

5.2 HS Panel Method

71

and 1

d

vii = - boil dx

(LiiuilSt)

(5.1.9)

Here the sign convention has been used that v is measured positive in the direction of the upward normal to the wake. Hence a jump ,6v in the component of velocity normal to the wake is required; it is given by d

dvi ° viu - uil =Piu 1 dx (Piuuiusu) + Ail d (Pituiiol) 1 dx

(5.1.10)

5.2 HS Panel Method We consider an airfoil at rest in an onset flow of velocity Vim. We assume that the airfoil is at an angle of attack, a (the angle between its chord line and the onset velocity), and that the upper and lower surfaces are given by functions Y,, (x) and Y(x), respectively. These functions can be defined analytically, or (as is often the case) by a set of (x, y) values of the airfoil coordinates. We denote the distance of any field point (x, y) from an arbitrary point, b, on the airfoil surface by r, as shown in Fig. 5.4. Let n also denote the unit vector normal to the airfoil surface and directed from the body into the fluid, and F a unit vector tangential to the surface, and assume that the inclination of F to the x-axis is given by 9. It follows from Fig. 5.4 that with a and j denoting unit vectors in the x- and y-directions, respectively,

n"=-sine?+cos0j t=cos9?+sin9j

(5.2.1)

If the airfoil contour is divided into a large number of small segments, ds, then we can write dx=cosOds dy = sin 9 ds

(5.2.2)

with ds calculated from ds = (dx)' + (dy) . We next assume that the airfoil geometry is represented by a finite number (N) of short straight-line elements called panels, defined by (N+1)(xj, yj) pairs called boundary points. It is customary to input the (x, y) coordinates starting at the lower surface trailing edge, proceeding clockwise around the airfoil, and ending back at the upper surface trailing edge. If we denote the boundary points by

(x1, yl), (x2, y2), ... , (XN, YN), (xN+l, YN+1)

(5.2.3)

then the pairs (xl, yl) and (xN+1, yN+l) are identical for a closed trailing edge (but not for an open trailing edge) and represent the trailing edge. It is customary to refer to the element between (xj, yj) and (xj+1, yj+i) as the j-th panel,

5. Hess-Smith Panel Method with Viscous Effects

72

source and vorticity distributions

trailing edge

V control points

y ° Yr(x)

boundary

Fig. 5.4. Panel representation of airfoil surface and notation for an airfoil at incidence a.

and to the midpoints of the panels as the control points. Note from Fig. 5.4 that as one traverses from the i-th boundary point to the (i + 1)-th boundary point, the airfoil body is on the right-hand side. This numbering sequence is consistent

with the common definition of the unit normal vector iii and unit tangential vector ti for all panel surfaces, i.e., fij is directed from the body into the fluid and ti from the i-th boundary point to the (i + 1)-th boundary point with its inclination to the x-axis given by Bi. In the HS panel method, the velocity 'Cl at any point (x, y) is represented by

V=U+v

(5.2.4)

where U is the velocity of the uniform flow at infinity

U=V.(cosaa+sincaj)

(5.2.5)

and v" is the disturbance field due to the body which is represented by two elementary flows corresponding to source and vortex flows. A source or vortex on the j-th panel causes an induced source velocity vs at (x, y) or an induced vortex

velocity ii at (x, y), respectively, and these are obtained by taking gradients of a potential source

s=2-Inr

(5.2.6)

Or = -2/8,

(5.2.7)

and a potential vortex

both centered at the origin, so that, with integrals applied to the airfoil surface, v(x) y) =

J

vsg3(s)dsj + f iYvrj(s)dsi

(5.2.8)

5.2 HS Panel Method

73

Here gjdsj is the source strength for the element dsj on the j-th panel. Similarly, Tjdsj is the vorticity strength for the element dsj on the same panel.

Each of the N panels is represented by similar sources and vortices distributed on the airfoil surface. The induced velocities in Eq. (5.2.8) satisfy the irrotationality condition and the boundary condition at infinity (5.2.9a)

y

(5.2.9b)

y

For uniqueness of the solutions, it is also necessary to specify the magnitude of the circulation around the body. To satisfy the boundary conditions on the body, which correspond to the requirement that the surface of the body is a streamline of the flow, that is, -0 = constant

or

=0

(5.2.10)

annormal, the sum of the sourceat the surface on which n is the direction of the induced and vorticity-induced velocities and freestrearn velocity is set to zero in the direction normal to the surface of each of the N panels. It is customary to choose the control points to numerically satisfy the requirement that the resultant flow is tangent to the surface. If the tangential and normal components of the total velocity at the control point of the i-th panel are denoted by (VL)i and (V')i, respectively, the flow tangency conditions are then satisfied at panel control points by requiring that the resultant velocity at each control point has

only (V%, and

(V')i=0 i=1,2,...,N

(5.2.11)

Thus, to solve the Laplace equation with this approach, at the i-th panel con-

trol point we compute the normal (Vn)i and tangential (Vt)i, (i = 1, 2, .... N) velocity components induced by the source and vorticity distributions on all panels, j (j = 1, 2, ... , N), including the i-th panel itself, and separately sum all the induced velocities for the normal and tangential components together with the freestaeam velocity components. The resulting expressions, which satisfy the

irrotationality condition, must also satisfy the boundary conditions discussed above. Before discussing this aspect of the problem, it is convenient to write Eq. (5.2.4) expressed in terms of its velocity components (V")i and (Vt)i by N N (Vn)i = E A?gj + > B injTj + Vo, sin(a - Bi) j=1

N

(Vt)i =

N

Bii

ALjgj +

j=1

(5.2.12a)

j=1

j=1

+ V cos(a - 8i)

(5.2.12b)

5. Hess-Smith Panel Method with Viscous Effects

74

where A'-, B , A(.,23Btu are known as influence coefficients, defined as the velocities induced at a control point (xzni, ynLi ); more specifically, A and A% denote the normal and tangential velocity components, respectively, induced at

the i-th panel control point by a unit strength source distribution on the j-th panel, and BR and B are those induced by unit strength vorticity distribution on the j-th panel. The influence coefficients are related to the airfoil geometry and the panel arrangement; they are given by the following expressions:

AZJ .

r 1 [sin(Oi - 8j) In rZ'j+l + cos(O - Bj )/3tij] Tij

i#j

2

i=j

1

AZ - =

1 27r

[si,(Oi - Oj)I3ij - cos(Oi - Oj) In r2'j+11 7ij

(5.2.13)

i0j

J

(5.2.14)

i=j

0

Binj =-A2j

(5.2.15)

Here

Tij+l =

[(x"ni - xj+1)2 + (ymi

- yj+1)2J 1/2

Ti,j = [(xmi - xj)2 + (ymi - yj)2J 1/2 xmi = 2 (-Ti + xj+1),

9i = tan-1 ,Qij =

yi+h - yi xi+1 - xi

tan-'

(Ymi - Yj+1

ymi =

2 (Yi + Yj+1)

8j = tan-1

-

tan-1

yj+1 - yj xj+1 - xj ymi - yj xmi - xj

(5.2.16)

xmi - xj+1 Regardless of the nature of qj (s) and 7 j (s), Eq. (5.2.12) satisfies the irrotationality condition and the boundary condition at infinity, Eq. (5.2.9). To satisfy the requirements given by Eq. (5.2.11) and the condition related to the circulation, it is necessary to adjust these functions. In the approach adopted by Hess and Smith [1], the source strength qj(s) is assumed to be constant over the j-th panel and is adjusted to give zero normal velocity over the airfoil, and the vorticity strength Trj is taken to be constant on all panels (Trj = r) and its single value is adjusted to satisfy the condition associated with the specification of circulation. Since the specification of the circulation renders the solution unique, a rational way to determine the solution is required. The best approach is to adjust the circulation to give the correct force on the body as determined by experiment. However, this requires advance knowledge of that force, and one of the principal aims of a flow calculation method is to calculate the force and not to take it as given. Thus, another criterion for determining circulation is needed.

5.2 HS Panel Method 711

For smooth bodies such as ellipses, the problem of rationally determining the circulation has yet to be solved. Such bodies have circulation associated with them, and resulting lift forces, but there is no rule for calculating these forces. If, on the other hand, we deal with an airfoil having a sharp trailing edge, we can apply the Kutta condition [5, 6]. It turns out that for every value of circulation except one, the inviscid velocity is infinite at the trailing edge. The Kutta condition states that the particular value of circulation that gives a finite

velocity at the trailing edge is the proper one to choose. This condition does not include bodies with nonsharp trailing edges and bodies on which the viscous effects have been simulated by, for example, surface blowing, as discussed (4]. Thus, the classical Kutta condition is of strictly limited validity. It is customary to apply a "Kutta condition" to bodies outside its narrow definition, but this is an approximation; nevertheless the calculations are often in close accord with experiment. In the panel method, the Kutta condition is indirectly applied by deducing

another property of the flow at the trailing edge that is a direct consequence of the finiteness of velocity; this property is used as "the Kutta condition." Properties that have been used in lieu of "the Kutta condition" in panel methods include the following:

(a) A streamline of the flow leaves the trailing edge along the bisector of the trailing-edge angle.

(b) Upper and lower displacement total velocities approach a common limit at the trailing edge. The limiting value is zero if the trailing-edge angle is nonzero.

(c) Source and/or vorticity strengths at the trailing edge must satisfy conditions to allow finite velocity.

Of the above, property (b) is more widely used. At first it may be thought that this property requires setting both the upper and lower surface velocities equal to zero. This gives two conditions, which cannot be satisfied by adjusting a single parameter. The most reasonable choice is to make these two total velocities in the downstream direction at the 1st and N-th panel control points equal so that the flow leaves the trailing edge smoothly. Since the normal velocity on the surface is zero according to Eq. (5.2.11)1 the magnitudes of the two tangential velocities at the trailing edge must be equal to each other, that is,

(Vt)N = -(Vt)1

(5.2.17)

Introducing the flow tangency condition, Eq. (5.2.11), into Eq. (5.2.12a) and noting that Tj = T, we get

N

N

Atjgj+Tl3;j+V,osin(a-B;)=0, i=1,2,...,N j=1

j=1

(5.2.18)

5. Hess-Smith Panel Method with Viscous Effects

76

In terms of the unknowns, qj (j = 1, 2, ... , N) and r, the Kutta condition of Eq. (5.2.17) and Eq. (5.2.18) for a system of algebraic equations which can be written in the following form,

Ax=b

(5.2.19)

Here A is a square matrix of order (N + 1), that is

A=

all

a12

...

a1j

a1N

a1,N+i

all

a22

...

a2j

a2N

a2,N+1

ail

ai2

...

aij

aiN

ai,N-I.1

(5.2.20)

aNl aN2 . . . aNj . . . aNN aN,N+l aN+1,1 aN+1,2 . . . aN+l,j . . . aN+1,N aN+1,N+1

and 9 = (q,. .. , q,.. . , qN, r)T and b = (b,.... ) bi, .... bN, bN+l)T with denoting the transpose. The elements of the coefficient matrix A follow from Eq. (5.2.18)

a

_ An

a3,

i = 1,2,...,N j= 1 2

N

(5 . 2. 21a)

N

ai,N+l=EBB, i=1,2,...,N

(5.2.21b)

j=1

A? are given by Eq. (5.2.13) and B i'j by Eq. (5.2.15). The relation in Eq. (5.2.20) follows from the definition of x' where r is essentially xN+l. To find aN+1,j (J = 1, ... , N) and aN+1,N+1 in the coefficient matrix A, we

use the Kutta condition and apply Eq. (5.2.17) to Eq. (5.2.12b) and, with r as a constant, we write the resulting expression as N

N

A'jgj j=1

01)

N

j=1

N

ANjgj +TEBNj +V... cos(a-9N) j=1

j=1

or as

N

N

E(Atlj + AttNj)gj + r E(Blj + BNj) j=1

j=1

(5.2.22b)

= -V,,v cos(a - 01) - Vmm cos(a - 0N) so that, aN+l,j = A t l j + A N j ,

j = 1, 2, ... , N

(5.2.23a)

5.3 Viscous Effects

77

N

aN+1,N+1 =

T(Bzj + BNj)

(5,2.23b)

j=1

where now Aij and ANj are computed from Eq. (5.2.14) and Bij and BNj from Eq. (5.2.15). The components of b again follow from Eqs. (5.2.18) and (5.2.21). From Eq. (5.2.18),

bi = -V0,,sin(a - Os),

i = 1, ... , N

(5.2.24a)

and from Eq. (5.2.22),

bN+1 = -Voo cos(a - 61) - Voo cos(a - 9N)

(5.2.24b)

With all the elements of aij determined from Eqs. (5.2.21) and (5.2.23) and the elements of b' from Eq. (5.2.24), the solution of Eq. (5.2.19) can be obtained by the Gaussian elimination method [7]. The elements of i are given by N+1

xi

box-1)

aii

- 1: j=i+1

i = N + 1.... ,1

(5.2.25)

where

a(k) xj

aij(k-1)

-

(k-1) aik (k-1) akj (k_1) akk

b(k= b(k-1) _ 2:2b (k-1) a i (k-1) k akk

k = 1,..., N

j=k+1,...,N+1 i=k+1,...,N+1

(5.2.26a)

0

aij = aij k = 1,...,N

i=k+1,...,N+1 b(o)

(5.2.26b)

= bi

5.3 Viscous Effects The viscous effects can be introduced into the panel method by (1) replacing the zero normal-velocity condition, Eq. (5.2.11), by a nonzero normal-velocity condition V(x) and by (2) satisfying the Kutta condition, Eq. (5.2.17), not on the surface of the airfoil trailing edge but at some distance away from the surface. Here it will be assumed that the nonzero normal-velocity distribution Vu(x)

along the surface of the airfoil and in its wake is known, together with the distance from the surface, say displacement thickness 6*, where the Kutta condition is to be satisfied. We now describe how these two new conditions can be incorporated into the panel method.

5. Hess-Smith Panel Method with Viscous Effects

78

To include the nonzero normal-velocity condition into the solution procedure, we write Eq. (5.2.18) as

N

N

A qj + r

B jqj + V. sin(a - 8z) = vztf(x,,,;)

(5.3.1)

j=1

j=1

To satisfy the Kutta condition at the normal distance 6* from the surface of the trailing edge, called the "off-body" Kutta condition, the total velocities at the N-th and first off-body control points are again required to be equal. Since the normal velocity component is not zero, we write the off-body Kutta condition at distance fi* as (5.3.2) ('V)N = -(V)1

where V is the total velocity at the two control points. The of body total velocities are computed from

V=

(V" 2 +

(Vt 2

)

)

V

=V"

Vn V

Vt

+Vt V = Vnsino + Vtcoso

(5.3.3)

where V" and Vt are computed by expressions identical to those given by Eqs. (5.2.12) at the two off-body control points, I = 1. I = N, that is, N

N

(V'), =

A jjgj + T E B1 j + V sin(a - 81) j=1

j=1

N

N

(Vt)1 = F Ajjgj +T E Bjj + Vo. cos(ct - 8j) j=1

(5.3.4a)

(5.3.4b)

j=1

and where

0 = tan-1[(V")I/(Vt)j]

(5.3.5)

With Eqs. (5.3.4), the expression for the total velocity given by Eq. (5.3.3) can be written as N

V = (Alj -sinO+Ajj cos(6)gj j=1 N

(5.3.6a)

+ T E (B72 sin O + Bt cos ¢) j=1

+ Vo. sin(o - 01) sin, + V... cos(a - 91) cos or as

N

N

j=1

j=1

VA'Ijgj+TB'' +V,,. cos(a-8j-¢) where

(5.3.6b)

79

5.3 Viscous Effects

B jj =Bjj sin O+Bjj cos q5

A'I j = A7 . sin O + Al j cos O,

I

=

An

2Tr

1

AIj =

2rr

[s in (Oi

[si n(O

- 8j ) In !1>L+1 + cos (8j - 8 j ) Q Ij ] rl,j ,

I - 9j) /31j - cos( Oi - Bj) In r1,j+1 r1,j

1

(5.3.7a)

(5 . 3. 7b)

(5 .3 . 7c)

J

BIj = -At, , Btj = A7j

(5.3.7d)

8''=81+0

(5.3.8)

If we define

then it can be shown that Eq. (5.3.6b) can be written as N

N

V = E A'Ijgj + T E BIj + V. cos(a - BI) j=1 j=1

(5.3.9)

where

1 [s i ni1o- 0 )131] - cos (8 - 8) I n rja J

(5 . 3. 10a)

r1,1+1 BI = 1 [sin(0 r - 8j ) In + cos (8'I - 8j ) O 1j]

(5 . 3 . 10b)

A'Ii

=

3

ri,j

27r

,

The off-body Kutta condition can now be expressed in a form similar to that of Eq. (5.2.22). Applying Eq. (5.3.2) to Eq. (5.3.9), we write the resulting expression as N

N

j=1

j=1

N

(5.3.11a)

N

Aljgj+7- >Blj+V,,,, cos(a-0i) j=1

j=1

or as

N

N

(Aij + Avj)gj + r E(Blj + Bvj) j=1

j=1

+V,, cos(a-0)+V,,,, cos(a-0'N) =0

(5.3.11b)

80

5. Hess-Smith Panel Method with Viscous Effects

5.4 Flowfield Calculation in the Wake As discussed in Section 4.2, the calculation of airfoils in incompressible viscous flows can be accomplished without taking into account the wake effect; that is, the viscous flow calculations are performed up to the trailing edge only and are not extended into the wake. This procedure, which may be sufficient at low to

moderate angles of attack without flow separation, is not sufficient at higher angles of attack, including post-stall flows. Additional changes are required in the panel method (and in the boundary-layer method), as discussed in this section.

The viscous wake calculations usually include a streamline issuing from the trailing edge of the airfoil. The computation of the location of this streamline is relatively simple if conformal mapping methods are used to determine the velocity field. In this case, the stream function V) is usually known, and because the airfoil surface is represented by ib(x, y) = const, the calculation of the wake

streamline amounts to tracing the curve after it leaves the airfoil. When the flowfield is computed by a panel method or by a finite-difference method, how-

ever, the results are known only at discrete points in the field in terms of the velocity components. In this case, the wake streamline is determined from the numerical integration of dy

v

dx

u

(5.4.1)

aft of the trailing edge with known initial conditions. However, some care is necessary in selecting the initial conditions, especially when the trailing edge is blunt. As a general rule, the initial direction of the streamline is given to a good approximation by the bisector of the trailing-edge angle of the airfoil. The panel method, which was modified only for an airfoil flow, now requires similar modifications to include the viscous effects in the wake which behaves as a distribution of sinks. It is divided into nwp panels along the dividing stream-

line with suction velocities or sink strengths q; = A vi (N + 1 < i < N + nwp), distributed on the wake panels and determined from boundary-layer solutions in the wake by Eq. (5.2.12). As before, off -body boundary points and "control" points are introduced at the intersections of the 6* surface with the normals through panel boundary points and panel control points, respectively. Summation of all the induced velocities, separately for the normal and tangential components and together with the freestream velocity components, produces (V")r and (Vt)I at I = 1, 2, ... , N - nwp. The wake velocity distribution, as the airfoil velocity distribution, is computed on the b*-surface, rather than on the dividing streamline. The total velocities are again computed from Eq. (5.3.3), with (WI), and (Vt)I from Eqs. (5.3.4), except that now

5.4 Flowfield Calculation in the Wake

81

N+nwp

(Vn), _

N

Aljgj +T

Brj +V,, sin(a - 01)

(5.4.2a)

j=1

j=1 N+nwp

(V')1=

N

Aljgj + r E Bjj +

0j)

(5.4.2b)

j=1

j=1

As before, the expression for the total velocity is written in the same form as Eq. (5.3.6a), except that now N+nwp

(Alj sin 0 + A1 cos c)qj

V= j=1

N

+TF (BIj sink+Blj cosh)

(5.4.3)

j=1

+V, sin(a-0j)sin0+V,,,, cos(a-01)cosct Bjj and Bj'j, are identical to those given by Eq. (5.3.7). where AIj Similarly, Eq. (5.3.9) with A13 and B1j given by Eq. (5.3.10) is N+nwp

N

A'Ijgj+T

V j=1

B'I1+Voocos(a-0'1)

(5.4.4)

j=1

and the Kutta condition given by Eqs. (5.3.11a) becomes N+nwp

N

A'Njgj+TE BNj+V, cos(a-B'N) j=1

j=1 N+nwp

(5.4.5a)

N

E Aljgj+rEBij+V... cos(a-0') j=1

j=1

or

N+nwp

N

(Aij + A'' )qj + r E(Blj + BNj) j=1

j=1

(5.4.5b)

+V,,. cos(a-01)+VVcos(a-0'N) =0

In computing the wake velocity distribution at distances 6* from the wake dividing streamline, the velocities in the upper wake are equal to those in the lower wake for a symmetrical airfoil at zero angle of attack. This is, however, not

the case if the airfoil is asymmetric or if the airfoil is at an angle of incidence. While the external velocities on the upper and lower surfaces at the trailing edge are equal to each other, they are not equal to each other in the wake region since the 8*-distribution in the upper wake is different from the 8*-distribution in the lower wake.

82

5. Hess-Smith Panel Method with Viscous Effects

5.5 Computer Program In this section we present a computer program for the panel method discussed in the previous sections. This program can be used interactively with the boundary-layer program of Chapter 4 so that, as discussed in detail in [4] and briefly in Section 5.6, more accurate solutions of inviscid and viscous flow equations can be obtained by including the viscous effects in the panel method of Section 5.2. The computer program of the panel method has five subroutines and MAIN, as described below.

5.5.1 MAIN MAIN contains the input information which comprises (1) the number of panels along the surface of the airfoil, NODTOT, and the number of panels in the wake, NW. The code is arranged so that it can be used for inviscid flows with and without viscous effects. For inviscid flows, NW is equal to zero. (2) The next input data also comprises airfoil coordinates normalized with respect to its chord c, s/c, y/c, [_m X(I),Y(I)J. If NW # 0, then it is necessary to specify the dimensionless displacement thickness 6*/c (-DLSP(I)), dimensionless blowing velocity u,w/uoa (= VNP(I)) distributions on the airfoil, as well as the wake coordinates XW(I), YW(I) of the dividing streamline, the dimensionless displacement thickness distribution on the upper wake DELW(I,1) and lower wake DELW(I,2) and velocity jump QW(I). It should be noted that all input data for wake includes values at the trailing edge. The input also includes angle of attack a (-ALPHA) and Mach number 1L1= (= FMACH). The panel slopes are calculated from Eq. (5.2.2). The subroutine COEF is called to compute A and bin Eq. (5.2.19) subroutine OBKUTA to calculate the of body Kutta condition, subroutine GAUSS to compute i, subroutine VPDIS to compute the velocity and pressure distributions, and subroutine CLCM to compute the airfoil characteristics corresponding to lift (CL) and pitching moment (CM) coefficients.

5.5.2 Subroutine COEF This subroutine calculates the elements aij of the coefficient matrix A from Eqs. (5.2.21) and (5.2.23) and the elements of b from Eq. (5.2.24). We note that N + 1 corresponds to KUTTA, and N to NODTOT

5.5.3 Subroutine OBKUTA This subroutine is used to calculate the body-off Kutta condition.

5.5 Computer Program

83

5.5.4 Subroutine GAUSS The solution of Eq. (5.2.19) is obtained with the Gauss elimination method described in Section 5.2.

5.5.5 Subroutine VPDIS Once x is determined by subroutine GAUSS so that source strengths qi (i = 1,2 , ... , N) and vorticity r on the airfoil surface are known, the tangential velocity component (Vt) at each control point can be calculated. Denoting qi with Q(I) and r with GAMMA, the tangential velocities (Vt)i are obtained with the help of Eq. (5.2.12b). This subroutine also determines the distributions of the dimensionless pressure coefficient Cp (m CP) defined by

_ 1 - Poo Cp _

(5 5 1a) .

.

(1/2)PV.2

which in terms of velocities can be written as

= 1-

(lit)2 (5.5.1b)

It is common to use panel methods for low Mach number flows by introducing compressibility corrections which depend upon the linearized form of the compressibility velocity potential equation and are based on the assumption of small perturbations and thin airfoils [5]. A simple correction formula for this purpose is the Karman-Tsien formula which uses the "tangent gas" approximation to simplify the compressible potential-flow equations. According to this formula, the effect of Mach number on the pressure coefficient is estimated from ,3+ [11100/(1 +)0)] (cp=/2)

(5.5.2)

and the corresponding velocities are computed from

V2=1+ g [1 - (1 + cgcP)1117]

(5.5.3)

Here ep; denotes the incompressible pressure coefficient. ?Vl. the freestream Mach number and /3 =

1 - N1.2,

c6 =

ry

2

11Y12o<.

c7 =

7

7- 1,

e8 = 27R1.2,

7 = 1.4 (5.5.4)

In this subroutine we also include this capability in the HS panel method.

5. Hess-Smith Panel Method with Viscous Effects

84

5.5.6 Subroutine CLCM The dimensionless pressure in the appropriate directions is integrated to compute the aerodynamic force and the coefficients for lift (CL) and pitching moment (CM) about the leading edge of the airfoil.

5.5.7 Subroutine VPDWK This subroutine calculates the total velocity and pressure coefficient at each control point along the upper and lower wakes separately. The normal and tangential components of the total velocities are computed from Eqs. (5.4.2a) and (5.4.2b).

References Ill Hess, J. L. and Smith, A. M. 0., "Calculation of Potential Flow About Arbitrary Bod[2] [3]

ies," Progress in Aerospace Sciences, Vol. 8, Pergamon Press, N.Y., 1966. Lighthill, M. J., "On Displacement Thickness," J. Fluid Mech., Vol. 4, p. 383, 1958. Lock, R. C. and Firmin, M. C. P., "Survey of Techniques for Estimating Viscous Effects in External Aerodynamics," Royal Aircraft Establishment Tech Memo, AERO 1900, 1981.

[4] [5] [6] [71

Cebeci, T., An Engineering Approach to the Calculation of Aerodynamic Flows, Horizons Publishing, Long Beach; CA and Springer, Heidelberg, Germany, 1999. Anderson, J., Aerodynamics, McGraw-Hill, NY, 1988.

Moran, J., An Introduction to Theoretical and Computational Aerodynamics, John Wiley, NY, 1984. Isaacson, E. and Keller, H. B., Analysis of Numerical Methods, John Wiley, NY, 1966.

Application of the Computer Program for the CS and k-e: Models to Other Higher-Order Turbulence Models

6.0 Introduction The computer program for CS and k-e models described in Chapter 3 can also be used to obtain the solution of the boundary-layer equations with other turbulence models, including flows with separation. Here we first consider nonseparating flows; then we discuss the application of the computer program of Chapter 3 to k-w and SST models in Section 6.1 and to the SA model in Section 6.2. In Section 6.3 we present a brief description of the extension of the k-e model to flows with separation.

6.1 Solution of the k-w and SST Model Equations The solution of the k-w model equations with the computer program of Chapter 3 is similar to the solution of the k-e model equations with wall functions. Again the k-w model equations, Eqs. (1.2.18), (1.3.10), and (1.3.11), are expressed in terms of Falkner-Skan variables. Since the SST model equations make use of the k-w model equations in the inner region and the k-E model equations in the outer region we express them, for the sake of compactness, in the following form in transformed variables

[(1+akEin)k']'-2mf'k+ml f k'+c+(f")2-Q*wk = x (flak - k'Lf ) (6.1.1) FT, ax ((1 + assn )w ]' + 2(1 - Fl)v ,2 Rx kIw' + mlw' f - (m - 1) f'w - 1aw2 + Rx(f")2

8w

x flax - w

\

ex J

(6.1.2)

6. Application of the Computer Program for the CS and k-e Models

86

where w and k are dimensionless, normalized by x/ue and 1/ue, respectively.

Equations (6.1.1) and (6.1.2) are the equations used in the SST model. To recover Wilcox's k-w model equations expressed in transformed variables, we let F1 = 1 and take ak = 0.5,

a,,, = 0.5,

Q = 0.075, K2

J9 *

rc=0.41,

=0.09,

(6.1.3)

ry=Q* - vW

In the SST model, the above constants are determined from the relation, Eq. (1.4.10),

0=F101+(1-F1)02

(1.4.10)

where the constant 01 is determined from Eq. (1.4.11) and the constant

'2

from Eq. (1.4.12). Ft is detemined from Eq. (1.4.6), where its arg1 given by Eq. (1.4.7) can be written as arg1 = min[max(A1, )2), A3]

f

In terms of transformed quantities, Al to A3 and CDk,, Al

=

y2w

A3 = 1

CD,,,, = max 2Qa 2 w

(6.1.5a)

w77 0.09

500u A2 =

are

R.

vrk-

0.09wy

(6.1.4)

_

500 (6.1.5b)

77 W

4AQu,2 k (6.1.5c)

CDk,,,y2

8k

20

=

2k

max( k'w',10-20)ij2

We first find the maximum of Al and A2 (say A4), then calculate the minimum of A4 and A3 and thus determine arg1 and F1. Once F1 is calculated, then the constants in Eqs. (6.1.1) and (6.1.2) are determined from the relation given by Eq. (1.4.10). For example,

o',,,=0.5F1+0.856(1-F1) ,6 = 0.0750F1 + 0.0828(1 - Fl) etc.

Next we determine the eddy viscosity distribution across the boundary layer.

In terms of transformed variables, Eq. (1.4.1) can be written as (v = f1, al = 0.31)

+ - E,n

E=

v

Rx

y

k

a1w > (F2

w a1k

I'vIF2

(6.1.6)

a1w < (F2

6.1 Solution of the k-w and SST Model Equations

87

where

n HH =

ueI.f"I

vx and F2 is determined from Eq. (1.4.2a) where arg2 is arg2 = max(2A1, A2)

(6.1.7)

In the SST model, once the constants are determined and the distribution of eddy viscosity is calculated, then Eqs. (6.1.1) and (6.1.2) are solved together

with the continuity and momentum equations; a new argl, arg2, F1 and F2, new constants and eddy viscosity distribution are determined. This procedure is repeated until convergence. It should be noted that, for F1 = 1, the whole region is the inner region governed by the k-w model equations. When F1 = 0, the whole region is governed by the k-E model equations. Before we discuss the solution procedure for the SST model equations, it is useful to point out that the structure of the solution algorithm for the k-s model equations with wall functions is almost identical to the one for the SST model equations. This means all the Aj, Bj, C, matrices have the same structure; the difference occurs in the definitions of the coefficients of the linearized momen-

tum, kinetic energy and rate-of-dissipation equations and in the definition of the boundary condition for w which occurs in the fourth row of A0-matrix. To describe the numerical method for the k-w model equations, we start with the kinetic energy equation, Eq. (6.1.1). and write it in the same form as Eq. (2.1.4) by defining Q and F by Q = ,0`wk,

F=O

(6.1.8)

The definition of P remains the same. Next we write Eq. (6.1.2) in the form

(b3w')'+Pi.-Ql+E=x(f'ax -W,af) +(m-1)f'w-miw'f

(6.1.9)

where

E = 2(l - Fl) j,,., Q1 =

R,x W

k'w'

6W2

(6.1.10)

Pi = 'YR=(f")2 With

w'=g Equation (6.1.9) can be written as

(b3q)'+P1-Q1+E=x(uaW --gaf)+(m-1)uw-mlgf TX a:r

(6.1.11)

6. Application of the Computer Program for the CS and k-c Models

88

A comparison of Eq. (6.1.11) with Eq. (2.2.4) shows that if we let e = w for notation purposes, then the coefficients of linearized specific dissipation rate equation are very similar to those given by Eqs. (3.3.8) and (3.3.9). Except for the definitions of Q and F in the kinetic energy-equation, Eq. (6.1.1), the coefficients of the linearized kinetic energy equation are identical to those given

by Eqs. (3.3.4) and (3.3.5). Appropriate changes then can be easily made to subroutine KECOEF in order to adopt the computer program of Chapter 3 to solve the kinetic energy and specific dissipation rate kinetic energy equations in the SST model. Of course, other changes also should be made, but these are not discussed here. A good understanding of the computer program for the k-e model equations is needed to make the necessary changes.

6.2 Solution of the SA Model Equations The solution of the SA model equations with the computer program of Chapter 3 is similar to the solution of the k-e and SST model equations. As before we again

express the equations in transformed variables. Since this model involves the solution of the continuity and momentum equations with the transport equation for eddy viscosity, Eq. (1.5.4), and since the continuity and momentum equations

are already transformed we only need to transform Eq. (1.5.4). In terms of the Falkner-Skan variables, this equation can be written as x

f' 8x - (vi

m1(lt )'f = cbi(I - ft2)9*vi +)2

rwlfw - K2 ft2)

(+t(yw)2 2 77

+ _[[(I + Vt )(vt ),), + Cb2[(Ut )')2]

(6.2.1)

We now define

g=(vt )', u=f', V=u'

(6.2.2)

and write the transformed equations (2.1.3) and (6.2.1) as

f' = u

(6.2.3a)

u' = v

(6.2.3b)

(-t+) ' = g

(6.2.3c)

\

(bv)' + mifv +m(1 - u2) = x ,uax - vax)

(6.2.3d)

6.2 Solution of the SA Model Equations

89

a[[(z±Yt

ft2}S*Yi

r+2

- (cwt fw - -2 ft2) 2 `+t(V*w )2 + "'19f = aYt x u- gafl -J ax

(6.2.3e)

ax

Using the numerical method discussed in Chapter 2, the above equations can be expressed in the following linearized form (note vt - v+ for convenience)

bfj - bfj-1 - 2(6uj + buj_1) = (rl)j

(6.2.4a)

buj - buj_l - 2, (bvj + bvj_l) = (r4)j_l

(6.2.4b)

6vj+ -

(6.2.4c)

2 (bgj + 6gj_1) = (r5)j-l

(sl)jSfj + (82)j6fj-1 + (83)j6uj + (S4)jSuj-1 + (s5)j6vj + (S6)jbvj_i + (87)j5vj+ + (S8)jbvv 1 = (T2)j

624d

(el)jbfj + (e2)j6fj-1 + (e3)j6uj + (e4)36u3_l + (e5)j6vj + (e6)jbvj_l + (e7)jbVj + (eg)jbvv 1 + (eg)jbgj

(6.2.4e)

+ (elo)j5gj-1 = (r3)j Here the coefficients of Eq. (6.2.4d) are

(191)j=&vj_1/2+ 21vj

(6.2.5a)

(82)j = &vj-1/2 + 2l vj-1

(6.2.5b)

(s3)j = -(rn + &)uj

(6.2.5c)

(34)j = -(m+a)uj_1

(6.2.5d)

(s5)j = bjhj l +

m1 2

fj +0.5x

aj-1/2 J

(86)j = -bj_1h- l + 21 fj_1 +0.5x MX 2 (S7)j =

(6.2.5e)

(XL

aavjhj

1

} j-1/2

(6.2.5f)

(6.2.5g)

ab

(s8)j = - av+) j-1 vj-lh3_1

(6.2.5h)

6. Application of the Computer Program for the CS and k-e Models

90

[1

a[bjvj

\82c2

l ax

(T2)j = x

-1/2 vj(80j-1/21

j-1/2

-bj-lyj-1 hj

+m1(fv)j-1/2+m(1-Uj-1/2)J

(6.2.6)

where with Al, A2i A3 given by Eq. (2.3.9),

m1=m21, a=1A3x 8f n (ax)j-1/2

(6.2.7a)

= Al fj -1l2 + A2 fj11/+ A3 fnj-1/2

-2

(6.2.7b)

The term a denotes the variation of b with respect to v+ and is given by 8 v+

= A, + v + 49f+

(6.2.8)

The coefficients of Eq. (6.2.4e) are m219j

el = a9j-1/2 + a9j-1/2 +

e2

2x

eg =

=

8x

8pr

8de

8v

_

8pr

1

e4=e3

j-1/2

(9v)j

1

2

j

8de

8v )j_1

2

e7

9j-1

1(aY+

e3

e6

ml

ev

j-1

_-auj_1/2+(8U+)j+2 (8U'+)j-(8Uf)j

1 (LVI)j

of,,

es = -auj-1/2 +

8v

e9 = 12x

j-1

+2 m1

R`Vi /2 +2 x

(8f 1j e10

8U+

8x

m1

-1/2

+

2

fj

8v+

- 1

all, +

fj-1 -

89 (af

lj

(a!A)j-I 9

6.2 Solution of the SA Model Equations

91

(/av+1J - mi(ft g)1/2 (r3)j = Uj-1/2X ( ax' x / j-1/2 - ga-1/2x (!Lf) ax j-1/2 de)j-1/2

(A. + pr "

Pr and de denote the diffusion, production and destruction terms,

Here

respectively; they are given by

fµ =

(1

Cz) [(1 + v )gj - (1 + "j 1)9j-ll hj

Q vj 1/2(gj - gj-i)hj l

v+ 2 [2(g2+ Uw)2fv2+ y+ 2 de

772 + lUw)2

[Gui fw

0

(v) z

Rx [1 - (

)2]

R:IvIv+I

-2 ft2

for wall boundary layers along the wake

The subscript on x, xte, denotes trailing edge and the term denotes the variation of the production term with respect to v+; its expression depends on how the variation is accounted for. In order for the solutions to converge quadratically, the variation should be performed with respect to v+ on all the variables in the production term which are functions of v+. () et al. are given by

apr

° (1 av = b, ade aim

av+

v+ if v > 0 Rx{-v+}if v<0

- ft.) (V+)2

-

j

aim

772 + (vt*»)2

a w lev l a J If

l+Cb2 9j a hj 1 + cb2 9j9j=i

(av+

hj

j-i

a fls

ag

93-1

1Cb293

2a

1 +Cb2 '+V+ a hj

hj

l cb2 9j - 9j-1 2 or hj Cb2 3-1/2

a

hj

v+ 2

-cbl (av+}

_2(77s

+ (vw.)2)

[_(L'+)2

fee + v+

a b,W)2) av+ + av+ +

+ cbl(1- ftz) 2v+

+ 2(g2 + (yw)2) fv2 1

I RTlvl

6. Application of the Computer Program for the CS and k-E Models

92

Ode

av+

2i/ afw _ cbi 0ft21 r (y+)2 K2 av+ + 772 + (v* )2 Cw, fw tj2 + (v* )2 Cwi &V+

cbt K2

f,,,, ft2 and their variations with respect to v

In the above equations, f,,,, and v+ are as follows

ft2 = Ct3 exp[-ct,(v+)2]

_ av+ - -2ct4 v+ ft2 49A2

fw= rr =

[772 + (L> )2]Y2

1+

)Cw3g1

"+v IzI + K2( 2 + (vw)2] fvz } -1

afw av Of,,,

av+

9fw 891 a(rr)

age a(rr) av = afw 8g1 a(rr) ag1 a(rr) av+

where

afw

_

a91 - [91(1+cws916)] -1+cu.2(6rr5-1)

a(gr)

z a(rr)

av

l

.ft2/

K2(2 + *2) Rxv+)

162(112 + xw) 1

ifu>0

-1

if v < 0

fv2j

6.2 Solution of the SA Model Equations

93

2

0(rr)

K2(772 + v2)

C9v+

IV+v'nx Ivl +

x [,2(7)2+YU2)

r.2(772 + vw) fv21

+v'-R-.

f9 v+

IvI+,2(772+v,)fv2

]-I

[vIvI+

+,2(772+v*w)

2

2(772+v2)f02

The boundary conditions are

f

u=v+=0 u=1, v+= e

77 =0, 77=77e,

In linearized form

(6.2.9)

Suo = bfo = bvo = 0

(6.2.10)

Su,J = 6V j+ = 0

Next, as before, Eqs. (6.2.4) and (6.2.6) are written in matrix-vector form, with five-dimensional vectors bj and rj for each value of j defined by Sfj

(r" j

6uj

¢j =

(r2)j

6Vj

,

rj =

(r3)j

6'j

(r4)j

6gj

(r5)j

(6.2.11)

The 5 x 5 matrices are

Ao=

-1

_h0

1

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

(82)j (S4)j (S6)j (S8)j

Bj =

Aj =

0 0

(e2)j (e4)j (e6)j (es)j (eio)j

0 0

(6.2.12a)

0 -1 -2 0 0 0 0 0 -1 -2

0

0

0

0

0

0

0

0

1

-h3/2

0

0

(81)j

(S3)j

(S5)j

(s7)j

0

(el)j

(e3)j

(e5)j

(e7)j

(e9)j

0

-1

-hj+1/2

0

0

0

0

0

1 < j < J

,

(6.2.12b)

0

-1 -hj+112

,

1<j<J-1

(6.2.12c)

6. Application of the Computer Program for the CS and k-s Models

94

Ci =

0

0

0

0

0

0

0

0

0

0

0 0 0

0

0

0

0 0

1

0

-hh+z/2 0

,

0<j<J-1

(6.2.12d)

1 -hh+1/2

0 1

-hJ/2

(si)J (33)J

0

0

(35)J (37)J

0 0

AJ = (el)j (33)J (35)J (87)J (s9)J 1 0 0 0 0

0

0

0

1

(6.2.12e)

0

6.3 Extension of k-E Model to Flows with Separation The extension of the k-E model with low Reynolds number effects to flows with separation is similar to the extension of the CS-model discussed in Chapter 4. We start with the wall boundary conditions given by Eq. (2.1.13) and edge boundary conditions given by Eq. (2.1.4). As in Section 4.0 we use the transformation given by Eq. (4.0.3) and again write the continuity and momentum equations in the form given in Eq. (4.0.4) and express the kinetic energy and rate of dissipation

equations in a form similar to those given by Eqs. (2.1.4) and (2.1.5). The difference between Eqs. (2.1.4), (2.1.5) and the new ones are due to the use of a different transformation. The ordering of the eight first-order equations is done as described below. For j = 0, with the first four equations corresponding to the wall boundary conditions, we write

fo = 0

(6.3.1a)

uo = 0

(6.3.1b)

ko = 0

(6.3.1c)

Eo = 0

(6.3.1d)

w'=0

(6.3.1e)

f' = u

(6.3.1f)

U! = v

(6.3.1g)

Momentum Eq. (4.1.2c)

(6.3.1h)

k' = s p' = q

(6.3.2a)

kinetic energy, similar to Eq. (2.2.3)

(6.3.2c)

For1<j<J-1 (6.3.2b)

6.3 Extension of k-e Model to Flows with Separation

95

rate of dissipation equation, similar to Eq. (2.2.4)

For j = J

(6.3.2d)

W, = 0

(6.3.2e)

f' = u

(6.3.2f)

u' = v

(6.3.2g)

Momentum Eq. (4.1.2c)

(6.3.3h)

k'=s

(6.3.4a)

p' = q

(6.3.4b)

kinetic energy, similar to Eq. (2.2.3)

(6.3.4c)

rate of dissipation equation, similar to Eq. (2.2.4)

uJ = wJ

(6.3.4d) (6.3.4e)

of j + (1 - \r7J)wj = 9i

xwJ(kj)+EJ=0 `

(6.3.4f)

(6.3.4g)

2

wJ(xdx -EJJ+Ce2kJ =0

(6.3.4h)

J

where

3

k = kup,

E=s

u° x

(6.3.5)

Next we write the difference approximations to the above equations with FLARE approximation applied to the momentum equation. The resulting nonlinear algebraic system is linearized with Newton's method. The solution of the linear system is written in the form of Eq. (2.3.15) and is solved by an algorithm similar to that given by subroutine KESOLV, subsection 3.5.5.

Companion Computer Programs

7.0 Introduction The accompanying CD-ROM contains four computer programs (see Section 7.5). The first one is for the CS and k-e models described in Chapter 3. It is applicable only for turbulent flows. The second computer program is an inverse boundary-layer method applicable to both laminar and turbulent flows, with and without flow separation, as described in Chapter 4. It employs the CS model. The third computer program is the Hess and Smith panel method for computing inviscid flow around airfoils. It includes viscous effects, so that it can be coupled to the inverse boundary-layer method of Chapter 4 to compute flows around airfoils in an interactive manner as shown in the fourth computer program in the accompanying CD-ROM. In this chapter we present sample calculations for each computer program. Section 7.1 contains five test cases for the k-e model. They correspond to zero pressure gradient, favorable and adverse pressure gradient flows. Section 7.2 contains sample calculations for the panel method of Chapter 5. Inviscid flow calculations corresponding to the NACA 0012 airfoil are performed

for several angles of attack to predict the performance characteristics of the airfoil as well as the external velocity distribution on the airfoil and in the wake.

Section 7.3 presents boundary-layer calculations for both laminar and turbulent flows with transition location specified. The external velocity distribution is obtained from the panel method. Typical boundary-layer parameters, such as dimensionless displacement thickness, b"/c, and local skin friction coefficient, cf, are calculated as a function of dimensionless chordwise distance x/c. Section 7.4 discusses the application of the inviscid and inverse boundary layer methods to the prediction of lift and drag coefficients of the NACA 0012

7. Companion Computer Programs

98

airfoil discussed in the previous two sections. Both computer programs are cou-

pled and special arrangements are introduced in order to calculate the airfoil characteristics at high angles of attack, including stall.

7.1 Test Cases for the CS and k-e Models There are five test cases for this computer program. They all use the notation employed in the Stanford Conference in 1968 [1]. For example, flow 1400 corresponds to a zero-pressure gradient flow. Flow 2100 has favorable, nearly-zero and adverse-pressure-gradient flow. All calculations are performed for Model = 1, 2, -1 and -2 (see subroutine 3.1.2). The predictions of four models with experimental data are given for c f, 6* and R© as a function of x in the accompanying CD-ROM. A summary of the freestream and initial conditions for each flow are summarized below.

1. Flow 1400: Zero-Pressure-Gradient Flow NXT = 61, ue/uref = 1.0, uref = 33 ms-1, c f = 3.17 x 10-3, Re = 3856, v = 1.5 x 10-5 m2s-1, REF = 1

2. Flow 2100: Favorable, Zero and Adverse-Pressure-Gradient Flow NXT = 81, uCef = 100 ft s-1, cf = 3.10 x 10-3, Re = 3770, v = 1.6 x 10-4 ft2s-1, REF = 1 3. Flow 1300: Accelerating Flow

NXT = 81, uref = 100ms-1,

cf=4.61x10-3, Re= 1010,v=1.54x10-5m2s 1,REF=1 4. Flow 2400: Relaxing Flow

NXT = 81, uc/uref = tabulated values, uref = 1, cf = 1.42 x 10-3, Re = 27,391, v = 1.55 x 10-4 m2s-1, REF = 1 5. Flow 2900: Boundary Layer Flow in a Diverging Channel NXT = 81, ue/uref = tabulated values, uref = 1, c f = 1.77 x 10-3, Re = 22,449.2, v = 1.57 x 10-4 ft2s'1, REF = 1 The input and output for each flow are given in tabular and graphical form in the accompanying CD-ROM. Figure 7.1 shows a comparison between the calculated results with experimental data for flow 1400 and Fig. 7.2 for flow 2100. The calculations for Model = 1, 2, -1 and -2 correspond to low Reynolds number flows with Huang-Lin and Chien models, zonal method and high Reynolds number flows, respectively.

7.2 Sample Calculations for the Panel Method Without Viscous Effects

99

o Bxp. dew

0

Baeas.G Xff Model

- CTIMiMMade1

---

CS-X8 Zare[bfed.l

- HR-XH Model

x.mmu

Fig. 7.1. Comparison of calculated results with the experimental data for flow 1400

7.2 Sample Calculations for the Panel Method Without Viscous Effects This test case is for a NACA 0012 symmetrical airfoil, with a maximum thickness of 0.12c: the pressure and external velocity distributions on its upper and lower surfaces are computed and its section characteristics determined using the panel method. Table 7.1, also given in the accompanying CD-ROM, defines the airfoil coordinates for 184 points in tabular form. This corresponds to NODTOT = 183. Note that the x/c and y/c values are read in starting on the lower surface trailing edge (TE), traversing clockwise around the nose of the airfoil to the upper surface TE. The calculations are performed for angles of attack of a = 0°, 8° and 16°. In identifying the upper and lower surfaces of the airfoil, it is necessary to determine the x/c locations where ue(- ue/u,,,) = 0. This location, called the stagnation point, is easy to determine since the ue values are positive for the upper surface and negative for the lower surface. In general it is sufficient to take the stagnation point to be the x/c location where the change of sign ue occurs. For higher accuracy, if desired, the stagnation point can be

7. Companion Computer Programs

100

oaw5 e

O A* dm

c k. gRM.&I

--- CSJBZ..lM dd 62000

HR-RE&

I

S. (n) 0.2500

50

is

20

2s

70

X,0

Fig. 7.2. Comparison of calculated results with the experimental data for flow 2100

determined by interpolation between the negative and positive values of 4be as a function of the surface distance along the airfoil. Figures 7.3 and 7.4 show the variation of the pressure coefficient Cp and external velocity "ue on the lower and upper surfaces of the airfoil as a function of x/c at three angles of attack starting from 0°. As expected, the results show that the pressure and external velocity distributions on both surfaces are identical to each other at a = 0°. With increasing incidence angle, the pressure peak moves

upstream on the upper surface and downstream on the lower surface. In the former case, with the pressure peak increasing in magnitude with increasing a, the extent of the flow deceleration increases on the upper surface and, we shall see in the following section, increases the region of flow separation on the airfoil. On the lower surface, on the other hand. the region of accelerated flow increases with incidence angle which leads to regions of more laminar flow than turbulent flow.

These results indicate that the use of inviscid flow theory becomes increasingly less accurate at higher angles of attack since, due to flow separation, the viscous effects neglected in the panel method become increasingly more im-

7.2 Sample Calculations for the Panel Method Without Viscous Effects

Table 7.1. Tabulated coordinates for the NACA 0012 airfoil 1.000000 .952400

.996060 .938140

.991140 .922150

.984290 .904490

.975520 .885240

.964880 .864460

.842250

.818680

.793860

.767880

.740840

.712850

.684010 .499930 .395971

.654460 .482486 .378964

.624290 .465056 .362108

.593630 .447665 .345420

.562610 .430339 .328917

.531330 .413103 .312618

.296550

.280736

.265190

.249928

.234965

.206040

.192102

.178538

.165366

.152604

.220333 .140264

.128362

.116914

.105932

.095430

.085421

.075921

.066938 .028319 .018656 .010980 .005310 .001667 .000078 .000617 .003232 .007894 .014568 .023245 .043183 .095432 .165370 .249926 .345423 .447669

.058480 .026575 .017237 .009895 .004561 .001260 .000012 .000910 .003868 .008867 .015874 .024883 .050557 .105933 .178541 .265191 .362109 .465060

.050557 .024883 .015874 .008867 .003868 .000910 .000012 .001260 .004561 .009895 .017237 .026575 .058480 .116916 .192106 .280738 .378968 .482490

.043180 .023245 .014568 .007894 .003232 .000617 .000078 .001667 .005310 .010980 .018656 .028319 .066938 .128364 .206043 .296555 .395977 .499930

.036365 .021660 .013316 .006977 .002653 .000380 .000201 .002132 .006116 .012120 .020130 .030116 .075922 .140266 .220334 .312622 .413111 .531330

.030116 .020130 .012120 .006116 .002132 .000201 .000380 .002653 .006977 .013316 .021660 .036366 .085424 .152607 .234966 .328918 .430347 .562610

.593630

.624290

.654460

.684010

.712850

.740840

.767880

.793860

.818680

.842250

.864460

.885240

.904490 .984290

.922150 .991130

.938140 .996060

.952400 1.000000

.964880

.975520

.000000

-.006760 -.020990 -.038180 -.052940 -.058179 -.060015

-.000570 -.008700 -.023800 -.040920 -.054006 -.058748 -.059934

-.001290 -.010850 -.026670 -.043590 -.055004 -.059216 -.059734

-.002270 -.013170 -.029590 -.046150 -.055926 -.059580 -.059412

-.003520 -.015650 -.032500 -.048590 -.056766 -.059836 -.058965

-.005020 -.018260 -.035350 -.050860 -.057516 -.059980 -.058401

-.057710

-.056893

-.055952

-.054892

-.053715

-.052415

-.050992 -.040134 -.027674 -.022849 -.017809 -.012567 -.007149 -.001565

-.049452 -.037974 -.026887 -.022023 -.016949 -.011676 -.006228 -.000626

-.047799 -.035719 -.026093 -.021192 -.016084 -.010783 -.005303

-.046040 -.033376 -.025292 -.020355 -.015213 -.009883 -.004373

-.044167 -.030954 -.024484 -.019512 -.014336 -.008977 -.003439

-.042199 -.028454 -.023670 -.018663 -.013454 -.008066 -.002503

.000626

.001565

.002503

.003439

.004373 .009883

.005303 .010783

.006228 .011676

.007149 .012567

.008066 .013454

.008977 .014336

101

7. Companion Computer Programs

102

Table 7.1. (continued) .015213

.016084

.016949

.017809

.018663

.019512

.020355

.021192

.022023

.022849

.023670

.024484

.025292

.026093

.026887

.027674

.028454

.030954

.033376

.035717

.037972

.040132

.042198

.046040

.047803

.049453

.050994

.052414

.044170 .053714

.054894

.055953

.056895

.057710

.058398

.058963

.059409

.059734

.059934

.060015

.059980

.059834

.059580

.059217

.058748

.058177

.057513

.056763

.055926

.055003

.054006

.052940

.050860

.048590

.046150

.043590

.040920

.038180

.035350

.032500

.029590

.026670

.023800

.020990

.018260

.015650

.013170

.010850

.008700

.006760

.005020

.003520

.002270

.001290

.000570

.000000

portant. This is indicated in Fig. 7.5, which shows the calculated inviscid lift coefficients for this airfoil together with the experimental data reported in [2] for chord Reynolds numbers, Rc (- uc c/v), of 3 x 106 and 6 x 106. As can be seen, the calculated inviscid flow results agree reasonably well with the measured values at low and modest angles of attack. With increasing angle of attack, the lift coefficient reaches a maximum, called the maximum lift coefficient, (C2)m., at an angle of attack, a, called the stall angle. After this angle of attack, while the experimental lift coefficients begin to decrease with increasing angle of attack,

the calculated lift coefficient, independent of Reynolds number, continuously increases with increasing a. The lift curve slope is not influenced by R. at low to modest angles of attack, but at higher angles of attack it is influenced by R,, dependent upon R. thus making (cg)1

Fig. 7.3. Distribution of pressure coefficient on the NACA

0012 airfoil at a = 0°, 8° and 16°.

7.3 Sample Calculations for the Inverse Boundary-Layer Program

103

--- a = 8 a=16 ................ ti

0.2

0.4

i

0.6

0.8

Fig. 7.4. Distribution of di1.0

x/c

mensionless external velocity on the NACA 0012 airfoil at a = 0°, 8° and 16°.

2.0 r

1.5

ce

1.0

0.5

Fig. 7.5. Comparison of calculated (solid lines) and ex0.0

0.0

5.0

10.0

a

15.0

20.0

perimental (symbols) lift coefficients for the NACA 0012 airfoil.

7.3 Sample Calculations for the Inverse Boundary-Layer Program This test case is again for the airfoil considered in the prevoius section. The boundary-layer calculations are performed only for the upper surface, for laminar and turbulent flows with transition location specified, at angles of attack of a = 40, 8°, 120, 140, 16° and 17°. The airfoil coordinates, x/c, y/c are used to calculate the surface distance. The calculations are done for a chord Reynolds number of 4 x 106.

In practice, it is also necessary to calculate the transition location. Two practical methods for this purpose are the Michel method and the en-method described, for example, in [3, 4]. The former is based on a empirical correlation between two Reynolds numbers based on momentum thickness, R9, and surface

7. Companion Computer Programs

104

distance R. It is given by

}

/

"per=1.174 1+ where

Re=

u",

22,4001

Rx,

0.46 )Rxtr

(7.3.1)

uex Rx=-

The accuracy of this method is comparable to the en-method at high Reynolds number flows on airfoils. The e'-method, which is based on the linear stability theory, is, however, a general method applicable to incompressible and compressible two- and three-dimensional flows. As discussed in [3, 4], for two-dimensional

flows at low Reynolds numbers, transition can occur inside separation bubble and can be predicted only by the en-method. For details, see [3, 4]. While the boundary-layer calculations with this program can be performed for standard and inverse problems, here they are performed for the standard problem, postponing the application of the inverse method to the following section.

The accompanying CD-ROM, under Section 7.3, presents the input and output of the calculations. Here we present a sample of them. The format of the inverse boundary-layer program is similar to the format of the interactive code and is discussed in the following section. Figure 7.6 shows the distribution of local skin friction coefficient, c f, and dimensionless displacement thickness, 6*/c, for several angles of attack. These results were obtained for the external velocity distribution provided by the panel method without viscous corrections. The boundary-layer calculations were performed in the inverse mode and several

sweeps on the airfoil and in its wake were made. As can be seen, at low or medium angles of attack, there is no flow separation on the airfoil corresponding to the vanishing of cf or v,,,. At higher angles, however, as expected, the flow

separates near the trailing edge and moves forward with increasing angle of

Fig. 7.6. Variation of (a) c1 and (b) 6'/c on the NACA 0012 airfoil and its wake at several angles of attack for R. = 4 x 106.

7.4 Sample Calculations with the Interactive Boundary-Layer Program

105

attack. It is interesting to note that at a = 16°, the flow separation occurs at x/c = 0.6 and at a = 17° at x/c = 0.37. As we shall see in the next section, interaction between inviscid and viscous results reduces the flow separation on the airfoil considerably. The results also show that, again as expected, transition location occurs very close to the stagnation point at higher angles of attack.

7.4 Sample Calculations with the Interactive Boundary-Layer Program A combination of an inviscid method with a boundary-layer method allows the inviscid and viscous flow calculations to be performed in an interactive way. Using an inverse boundary-layer method allows similar calculations to be performed for flows including separation. Before we present sample calculations with the interactive boundary-layer program, it is first useful to discuss the computational strategy in this program. For a specified angle of attack a and airfoil geometry (x/c, y/c), the calculations

are first initiated with the panel method in order to calculate the external velocity distribution and the lift coefficient. The external velocity distribution is then input to the inverse boundary-layer program in which, after identifying the airfoil stagnation point, the calculations are performed separately for the upper and lower surfaces of the airfoil and in the wake. The calculations involve several sweeps on the airfoil, one sweep corresponding to boundary-layer calculations

which start at the stagnation point and end at some specified c-location in the wake. In sweeping through the boundary-layer, the right-hand side of Eq. (4.0.6) uses the values of 5* from the previous sweep when j > i and the values from the current sweep when j < i. Thus, at each e-station the right-hand side of Eq. (4.0.6) provides a prescribed value for the linear combination of

and 5*(ci). After convergence of the Newton iterations at each station, the summations of Eq. (4.0.6) are updated for the next -station. Note that the Hilbert integral coefficients Cij discussed in subsection 4.2.3 are computed and stored at the start of the boundary-layer calculations.

At the completion of the boundary-layer sweeps on the airfoil and in the wake, boundary-layer solutions are available on the airfoil and in the wake. The blowing velocity on the airfoil viw [see Eqs. (5.1.8) and (5.1.9)] and a jump in the normal velocity component dvi in the wake (see Eq. (5.1.10)], for which an incompressible flow are d

Vi. = dx ('uit55 Qvi =

)

d d dx (uiusu) + dx (u:t5t)

(7.4.2a)

(7.4.2b)

are calculated and are used to obtain a new distribution of external velocity uei(x) from the inviscid method. As before, the onset of transition location is

7. Companion Computer Programs

106

determined from the laminar flow solutions and the boundary-layer calculations are performed on the upper and lower surfaces of the airfoil and in the wake by making several specified sweeps. This sequence of calculations is repeated for the whole flowfield until convergence is achieved. The format of the input to this interactive boundary-layer (inviscid/viscous) program is similar to the input required for the inverse boundary-layer described in subsection 4.2.1. The code is arranged in such a way that it is only necessary

to read in the airfoil geometry, the angles of attack to be calculated, Mach number and chord Reynolds number. The rest of the input is done internally. We now present sample calculations for the NACA 0012 airfoil for Reynolds numbers corresponding to 3 x 106. In this case, transition locations are calculated with Michel's formula. The calculations and the results are given in the accompanying CD-ROM. Lift, cl, drag, cd, pitching moment, cm, coefficients for R. = 3 x 106 are shown

in Table 7.2 for a = 2° to 16.5° and M,,, = 0.1 together with lift coefficients calculated with the panel method. As can be seen, while at low and modest angles of attack, the inviscid lift, cll,,, and viscous lift, cl,,, coefficients agree reasonably well, at higher angles of attack, as expected, they differ from each other. Figure 7.7 shows a comparison between the calculated and experimental values of lift and drag coefficients. The agreement is good and the stall angle is reasonably well predicted. For additional comparisons with experimental data, see [3].

To describe the input and output of the computer program, we now present additional calculations for the same airfoil, this time for R, = 4 x 106. The input to the IBL program (Fig. 7.8) includes airfoil geometry and/or

the number of angles of attack (N), the freestream Mach number, M , and Table 7.2. Results for the NACA 0012 airfoil ar R. = 3 x 106, M°° = 0.1. a

Clin

Clvi

Cd

Cml°

Cmvi

2.00000

0.24261

0.21099

0.00586

4.00000

0.48508

6.00000

0.72727

0.42567 0.64337

0.00610 0.00749

-0.06326 -0.12622 -0.18857

-0.04971 -0.10099 -0.15325

8.00000

0.96908

0.86241

0.00955

-0.25003

-0.20621

10.00000

1.21041

1.07109

0.01178

-0.31029

-0.25434

12.00000

1.45120

1.26253

0.01498

-0.36907

-0.29536

13.00000

1.57138

1.34396

0.01658

14.00000

1.69142

1.40836

0.01892

15.00000

1.81133

1.44754

0.02181

15.50000

1.93110

1.45653

0.02366

-0.39782 -0.42609 -0.45385 -0.48107

-0.31005 -0.31856 -0.31873 -0.31636

16.00000 16.50000

1.99094

1.45811 1.44226

0.02592 0.02837

-0.49446

-0.31261 -0.30540

7.4 Sample Calculations with the Interactive Boundary-Layer Program

107

0.040

l

0.5

a

1.0

1.5

Cl (b)

(a)

Fig. 7.7. Comparison between calculated (solid lines) and experimental values (symbols) of. (a) ct vs a, and (b) cd vs cr. NACA 0012 airfoil at R. = 3 x 10".

the Reynolds number, R. The input file in the sample calculations contains the NACA 0012 airfoil coordinates which are specified by choosing either M1M4 or M1M4INP. The first choice contains only the airfoil geometry and does not contain either the angles of attack, Mach number or Reynolds number. The second choice contains airfoil geometry, angles of attack, Mach number and Reynolds number. If the first one is chosen, then it is necessary to specify N, MM and R. For example if N = 5, then the angles of attack can be, say, 0°, 4°, 6°, 8° and 9°. Of course, these angles of attack as well as N can be changed. Then the calculations are started by specifying Mc,, and R. Figure 7.8 shows a sequence of the screens used for input.

r* Edt IrvA Fle

8L Output

ParrelOutputII

Output Summary

Numbs of ante at attack 1' 1 stAttack Angte

Mach Number

I

Reynolds Number

f

0.0

t ang e e auaek Ii

istAttackArgle

0.0

Fig. 7.8. Input format.

Mach Nmber

7. Companion Computer Programs

108

.JJ Look

in:ti Panel IBL Text n0012.txt

r

File name:

files of typei

Open Cancel

Next File (".tx@

r Open as read-only

.11

JJJ

He Edit Computation

iL

Input File

BLOu1put

Panel OtkwtFD:\PaneUBL-Teid\nOM2PanelOtd.txt Number of angle of attack 114

Mach Ntaiiber

0:1Panel_0L Text1n00128L0ut.txl

Output Summary " 0:1Panel_IBL_Text1n0012SummaryOut.hd

Ql

ReynoldoNumber 13000000

l st angle of attack r 2th angle of attack 14 3th ogle of attack 4th angle of attack

la

5th angle of attack

10

0th angle of attack

12

7th angle of attack

13

0th angle of attack

14

91h angle of attack

15

101h angle of attack

15.5

11thangeofattack

16

as

121h angle of attack

13th angle of attack

17

14th angle of attack

17.5

Fig. 7.8. (continued)

Figure 7.9 shows the screen for starting the calculations and Fig. 7.10 shows the screen for the format of the output and the variation of lift coefficient with angle of attack. Other plots to include cd vs a, c11 vs a and cd vs cl can also be obtained as shown in Fig. 7.11. Finally, the screen in Fig. 7.12 shows that one can copy the plot to the Microsoft Word file.

7.4 Sample Calculations with the Interactive Boundary-Layer Program

109

1X1 Fde

Edit i Computation

InpU fqk

BLOutpul

Scnr,r7Ut. ra

Number of angle of attack 14

Mach Number

D:1Paxl IBL Text1n00128L0u11xt

OWlxd Sunoary D:1Pae1_IBL Tet1n0012SurrmaryOut.txl

Panel DNput p_1Panel-IBL Text\nOO12Panequt.txl

Re

0.1

ids Number

3000000

Id angle of attack 2th angle of attack

1-

3th angle of attack J6 4th angle of attack

5ttangleol attack

f6

_

10

6th angle of attack

12

7th angle of attack

13

8th angle of attack

14

9th angled attack

15

10th angled attack 15.5

l

angle

eltadk

0.10

16

0.05

12th angleof of attack 11th

165

13th angie of attack

17

-0.05

14th angle of attack

17.5

.0.10

> oA0 x

Fig. 7.9. Beginning of calculations. rant Rte

Edk

to

Computatbn f Slww BL Output

Imu Pie

D:1Panel-IBLTOM1nO012BLAul.lxl

Draw angle vs Cd

Pond 0utpuf 0 Panel_

Draw angle vs Cm

Number of angle of a6ac

Draw Cl vs Cd

ld angle c attack r

Oraw Cl vs Cm

OutputSumtary

2th angle of attack r-

D:tPanelIBL_Te0xn0012Summat,Ou1tct Reyrwlds Number

01

Mach - 0.1

Reynolds

3000000

300000

3th angle of attack re-

4th angle of attack r5th engie of attack

10

6th angle of attack

12

7th angle of attack

13

6th angle of attack

14

9th agile of attack

15

10th angle of attack

15.5

1.6

1.4

1.2

11th angle of aback 16 12th angle of attack

16.5

13th angle of attack

17

1.0

0 0.6

OR

OA

02 2

4

6

6

10

Angle of Made

Fig. 7.10. Output format.

12

14

16

18,

7. Companion Computer Programs

110

0030

.005 1

0025

.0.03

00.0

.0±5

1

1

r

s 470 001$

475 0010

420 coo 0

10

12

n0

435 PCSCY A&

Fig. 7.11. Calculated results for the NACA 0012 airfoil, R = 4 x 106, Ma, = 0.1. (a) Cd vs a, (b) Cm VS or.

Show ..

Fie 1Edt computation

BLOutpta

Nu nber of angle of attack 13

Mach Number

D:\Panel IBL Text\n0012BL0uttxt

Output Siamrary

Panel Output D:\Panel IBLText\nO012ParrelOut.txt

D:\Panel IBL Text\n0012Summatt0uttxt Reynodu Number

0.1

13000000

15t angle of attack

2th angle of attack 4-

5th angle of attack

r-

6th angle of attack

12

71h an g le of attack

13

8th angle of attack

1d

9th angle of attack

15

1 Olh angle of attack

15.5

llthangleofattack

16

12th angle of attack

16.5

1301 angle of attack

17

4th angle of attack attack 4th angle of

each

r

0.1

Reynolds 3000000

0.035

10

0.030

0.025

0.020

0.015

0.010

0.0D5

0.2

0.4

0.6

1.0

0.8

ci

Fig. 7.12. Instruction for copying plots,

12

1.4

1.6

7.4 Sample Calculations with the Interactive Boundary-Layer Program

111

2.5

21 1.5

C1

0.5 d

040

(a)

0.005

0.004 0.05

0.003

5'k 0.06

CF 0.002

............. .

0.04

0.001

0.02

0

.0.001

(c)

We

1

2

(d)

3

4

5

6

ah

Fig. 7.13. Comparison of results between the inverse boundary-layrr method and the interactive method. (a) q vs a, (b) cd vs a, (c) cf vs x/c, (d) b'/c vs x/c.

Figure 7.13 shows a comparison between the results of the previous section where the inviscid flow calculations did not include viscous effects, and the results of this section which include viscous effects in the panel method. Figures

7.13a and 7.13b show the strong influence of viscosity on c1 and cd. Figure 7.13c shows that with interaction, the extent of flow separation on the airfoil decreases. For example at ce = 17°, without viscous effects in the panel method,

the flow separation occurs around x/c - 0.37. With interaction, it occurs at x/c = 0.62. Similarly, with interaction, the peak in 6*/c (Fig. 7.13d) decreases and is the reason for less flow separation on the airfoil.

7. Companion Computer Programs

112

7.5 Computer Programs in the CD-ROM The CD-ROM accompanying this book contains both source and executable computer programs and test cases. They are listed below.

CS and k-e Program Compile kemodel. for (Boundary-Layer Method with k-e Model) and generate PC executable file kemodel. exe. Test Case 1300:

a) Huang-Lin Model Input data:

1300_linp.txt

Output data:1300_1out.txt or 1300_loutS.pdf

b) Chien Model Input data:

1300_2inp.txt

Output data:1300_2out.txt or 1300_2outS.pdf

c) CS-KE Zonal Model Input data:

1300_mlinp.txt

Output data:1300_mlout.txt or 1300 mloutS.pdf

d) Standard High-Reynolds Number KE Model Input data:

1300_m2inp.txt

Output data: 1300m2out.txt or 1300 m2outS.pdf

Results Summary: 1300_out.pdf Test Case 1400:

a) Huang-Lin Model Input data:

1400_linp.txt

Output data:1400_lout.txt or 1400_loutS.pdf

b) Chien Model Input data:

1400_2inp.txt

Output data: 1400_2out. txt or 1400_2outS.pdf c) CS-KE Zonal Model Input data:

1400 miinp.txt

Output data: 1400mlout.txt or 1400 mloutS.pdf d) Standard High-Reynolds Number KE Model Input data.

1400_m2lnp.txt

Output data: 1400m2out.txt or 1400 m2outS.pdf

Results Summary: 1400_out. pdf

7.5 Computer Programs in the CD-ROM

Test Case 2100:

a) Huang-Lin Model Input data:

2100_linp.txt

Output data:2100_lout.txt or 2100_loutS.pdf b) Chien Model Input data:

2100_2inp.txt

Output data: 2100_2out.txt or2100_2outS.pdf

c) CS-KE Zonal Model Input data: 2100mlinp.txt Output data: 2100_mlout.txt or 2100_mloutS.pdf

d) Standard High-Reynolds Number KE Model Input data:

2100_m2inp.txt

Output data: 2100_m2out. txt or 2100_m2outS. pdf Results Summary: 2100_out.pdf Test Case 2400:

a) Huang-Lin Model Input data: 2400_ iinp . txt Output data: 2400_lout.txt or 2400_loutS.pdf

b) Chien Model Input data:

2400_2inp.txt

Output data: 2400_2out. txt or 2400_2outS. pdf c) CS-KE Zonal Model Input data:

2400_mlinp.txt

Output data: 2400_miout.txt or 2400_mloutS.pdf d) Standard High-Reynolds Number KE Model Input data:

2400_m2inp.txt

Output data: 2400_m2out.txt or 2400_m2outS.pdf

Results Summary: 2400_out.pdf Test Case 2900:

a) Huang-Lin Model Input data:

2900_iinp.txt

Output data: 2900_lout.txt or 2900_ioutS.pdf b) Chien Model Input data:

2900_2inp.txt

Output data: 2900_2out.txt or2900_2outS.pdf

c) CS-KE Zonal Model Input data:

2900 mlinp.txt

Output data: 2900mlout.txt or 2900_mloutS.pdf

113

7. Companion Computer Programs

114

d) Standard High-Reynolds Number KE Model Input data: 2900 m2inp.txt Output data: 2900_m2out. txt or 2900_m2outS. pdf Results Summary: 2900_out.pdf

Panel Method (HSPM) Compile pane13. for and generate PC executable file panel3. exe. Test Cases NACA0012 Airfoil, M = 0.1 1.

a = 0° angle of attack. Input file: deg_00. inp Output file: deg_00. out and deg_00. bdf

2.

a = 8° angle of attack.

Input file: deg_08. inp Output file: deg_08.out and deg_08.bdf 3.

a = 16° angle of attack. Input file: deg_ 16 . inp Output file: deg_16.out and deg_16.bdf

Inverse Boundary-Layer Method (IBLM) Compile ibim.for (Inverse Boundary-Layer Method) and generate PC executable file iblm.exe. Test Case NACA0012 Airfoil, M = 0.1 and RL = 4.OM 1. a = 0° angle of attack. Input file:

n0012aOinp.txt

Output file: n0012a0out. txt or n0012aOout . pdf 2.

a = 4° angle of attack. Input file:

Output file:

3.

nOO12a4inp.txt n0012a4out.txt or nOO12a4out.pdf

a = 8° angle of attack. Input file:

Output file:

n0012a8inp.txt n0012a8out.txt or nOO12a8out.pdf

4. a = 12° angle of attack. Input file:

n0012al2inp.txt

Output file: nOO12al2out. txt or nOO12a12out.pdf

5. a = 14° angle of attack. Input file:

Output file:

n0012a14inp.txt nOO12a14out.txt or nOO12a14out.pdf

7.5 Computer Programs in the CD-ROM

6.

a = 16° angle of attack. Input file:

Output file:

7.

115

n0012a16inp.txt n0012a16out.txt or nOOl2al6out.pdf

a = 17° angle of attack. Input file:

nOOl2al7inp.txt

Output file: n0012a17out. txt or nOOl2al7out.pdf The results summary nOO12_Out.pdf

Interactive Boundary-Layer (IBL) Program Compile Interactive Boundary-Layer Program IBL. for (Panel and Inverse Boundary-Layer Methods) and generate PC DOS executable file IBL.exe. IBL PC Window Version is also available from this CD IBL_Window_Package.zip.

Click to see installation instruction. Test Case for NACA 0012 Airfoil, Mach = 0.10

a) RL = 3000000.0 Input data:

mlrm3.inp

Output data: Panel Method mirm3Panel . out IBL Method mlrm3ibl. out Summary mirm3Summary.out and mlrm3Sur.mary.bdf b) RL = 4000000.0 Input data:

mirm4.inp

Output data: Panel Method mlrm4Panel. out IBL Method mirm4ibl. out Summary mirm4Summary.out and m1rm4Summary.bdf

The summary mlrm4SummaryComp.pdf

7. Companion Computer Programs

116

References [1)

(2) [31

[41

Coles, D. and Hirst, E. A.: Computation of Turbulent Boundary Layers -1968 AFOSRIFP-Stanford Conference, vol. 2, Thermosciences Division, Stanford University, Stanford, Calif., 1969. Abbott, J. H. and von Doenhoff, A. E.: Theory of Wing Sections. Dover, 1959. Cebeci, T.: An Engineering Approach to the Calculation of Aerodynamic Flows. Horizons Publishing, Long Beach, CA and Springer-Verlag, Heidelberg, Germany, 1999. Cebeci, T. and Cousteix, J.: Modeling and Computation of Boundary-Layer Flows, Horizons Publishing, Long Beach, CA and Springer-Verlag, Heidelberg, Germany, 1998.

Subject Index

Basic tools 54 Box method 21 - block-elimination method 30 - difference equations 23 - solution of the difference equations 25 Boundary-layer equations 2 - boundary conditions 3, 5

- continuity 2 - initial conditions 37, 56 - laminar flows 2 - momentum 2, 3 - turbulent flows 2, 3

- FLARE 58

- numerical method 58 - transformed equations 18, 57 Inverse problem 55 k-e model 5, 46 - solution without wall functions 31 solution with wall functions 33 k-w model 9

Cebeci-Sinith (CS) model 2, 40 Computer program - eddy-viscosity formulation of Cebeci and Smith 2, 40 - k-c model 5 - initial velocity profile

- computer program 62 - discretization approximation to the Hilbert integral 64 - Flugge-Lotz Reyhner approximation 58

37

- subroutine EDDY 43 - subroutine IVPT 37

- solution procedure

86

Law of the wall 5 Local skin-friction coefficient 22 Logarithmic law of the wall 5 Low-Reynolds-number effect 5, 7 Matrix solver algorithm

Computer program for flows with separation 42, 103, 105

Michel's method 103 Mixing length concept

Eddy viscosity concept

NACA 0012 airfoil

I

30, 53 1

99, 103, 104, 106. 107,

111

Friction velocity

3, 6, 20, 21

Hess-Smith panel method (HS) HS (Hess-Smith panel method)

- coordinates 101 Numerical solution of the boundary-layer equations 22-29 - block-elimination method 30 - Falkner-Skan transformation 17 - finite-difference approximation 23 - linear system 25, 29

67 67

Interaction problem - displacement thickness 55, 69 - interaction laws 55, 56 - interactive boundary-layer 67

- Newton's method

- Kutta condition 77, 78 - matching conditions in the wake 70 - mathematical models 68 - transpiration velocity

69

Interactive boundary-layer theory

Intermittency factor 3, 4 Inverse boundary-layer method

19

- numerical formulation 19, 23, 27, 58 - similarity variables 17 One-equation models - SA model 13

67

55

Panel method 67 - computer program

82

Subject Index

118

- tangent gas approximation

- viscous effects 77 - wake 77 Prediction of transition

- e"-method

Subroutine OUTPUT 40

83

Subroutine SOLV3 43 - block-elimination method

103

- empirical correlations

- Michel's method

30, 43

- Newton's method 19 Subroutine VPDIS 83 Subroutine VPDWK 84

103 104

Subroutines EDDY, GAMCAL, CALFA

103

45

Reynolds stress

Subroutines KECOEF, KEPARM, KEDEF

1, 2

and KEDAMP 46

SA model

- solution procedure Sample calculations

-panel method

Thompson's formula 6 Transformed equations 17, 57

88

- stream function 17, 57

99

- inverse boundary-layer program

103

- interactive boundary-layer program 105

Separation 55 Separation point 55 Singularity of boundary layer equations 55

- coupling 55 - singular behavior of boundary-layer equations 55 Singularity of the separation point 55 Solution algorithm 52 Solution procedure

- k-c model to flows with separation k-w model

86 - SA model 88 - SST model 86

SST model 10 Subroutine CLCM

94

Transformed variables 17, 86 Transport equations 5, 9, 10, 12, 13 - eddy viscosity 13 - momentum 2, 3 -: rate of dissipation 5, 7 - specific dissipation 9, 12 - turbulence kinetic energy 5, 7, 12 Turbulence models 1-13 - algebraic mixing length 1, 2 - eddy viscosity models 1-3 - one-equation models 13 - two-equation models 5, 9, 10 - 2ero-equation models 1-3 Two-equation models 5, 9, 10 -- k-e model 5, 46 - k-w model 9 - two-layer method 22 - wall functions 7

84

Subroutine COEF 82 Subroutine COEFTR 41 Subroutine GAUSS 83

Viscous-inviscid coupling 55, 69 - blowing velocity 69-71

- Hilbert integral 56

Subroutine GRID 39

Subroutine GROWTH 39 Subroutine HIC 64 Subroutine INPUT 36, 63 Subroutine IVPL 64 Subroutine IVPT 37 Subroutine KEINITG 52 Subroutine KEINITK 51 Subroutine KESOLV 52 Subroutine KEWALL 52 Subroutine MAIN 35, 63. 82

Subroutine OBKUTA 82

Wake 70, 80 Wall functions Wall shear 4

5

Zero-equation models

1-3

- mixing length 1, 2 Zonal method 22 - inner region 22 - interface between inner and outer region 27

- outer region 28

(l I1t t.1

Turhulencc %lod(Is and Their lpplication

OM With CO-ROM

I he (:1)-ROM

Required: Operating Systetn:

Mn 95, Win 98. Win itMI:.

Mn N14.0. Win 2000. Win XP or UNIX Web browser: Internet Explorer 5.5 (or higher)

ur NeLscape 4.7 (or higher) Adobe Arrohat Ileatler4.0 for higher) UNIX I OItTI(AN compller if using UNIX. Since PC executable Tiles

are on the CI)-ROM. it do" not require PC compiler

In this book, after it brief review of the more popular turbulence models, we present and discuss accurate and efficient

numerical methods for solving the boundary-layer equations with turbulence models based on algebraic formulas (mixing length, eddy viscosity) or partial-differential transport equations. A computer program employing the CebtxiSmith model and the k-r model for obtaining the solution of two-dimensional incompressible turbulent flows without separation is discussed in detail and Is presented in the accompanying CD-ROM.

The numerical solution of the boundary-layer equations with flow separation Is also discussed and an interactive

boundary-layer method developed by the author is described in detail. A computer program for obtaining solutions on airfoils and wakes for a wide range of angles of attack. including stall, is discussed and presented in the accompanying CD-ROM. In many respect,". this book can he

seen its complimentary and supplementary to the author's Analysis of Turbulent Flows.

ISBN a-966S461-6.8

ISBN 3-540-4D288.8